ATLANTA — Democrats moved a major step closer to gaining control of the U.S. Senate on Wednesday morning as Georgia voters elected the Rev. Raphael Warnock, the pastor at the storied Ebenezer Baptist Church, in a hard-fought runoff contest that became roiled by President Trump’s false claims of voter fraud in the state.Mr. Warnock’s victory over the Republican incumbent, Kelly Loeffler, represented a landmark breakthrough for African-Americans in politics as well as for Georgia: He became the first Black Democrat elected to the Senate from the South. For Democrats to take the Senate, which is crucial to enacting President-elect Joseph R. Biden Jr.’s first-term agenda, they also need to win Georgia’s other Senate runoff held on Tuesday. The votes were still being counted in that race between the Republican candidate, David Perdue, and his Democratic rival, Jon Ossoff. But turnout in rural, overwhelmingly white counties where Republicans needed a strong showing was lagging without Mr. Trump on the ballot, and many of Georgia’s heavily Black localities saw turnout levels that neared those of the presidential race in November.“May my story be an inspiration to some young person who is trying to grasp and grab hold to the American dream,” Mr. Warnock, who grew up in poverty, said in an online video just before 1 a.m. Wednesday. Invoking his mother, he said: “Because this is America, the 82-year-old hands that used to pick somebody else’s cotton went to the polls and picked her youngest son to be a United States Senator.”With about 97 percent of the vote counted in the early hours of Wednesday, Mr. Warnock held a lead of roughly 47,000 votes over Ms. Loeffler, after surging ahead late in Tuesday night when heavily Democratic DeKalb County reported a trove of ballots. While Mr. Warnock’s win was a major gain for his party — he is the first Democrat to be elected to the Senate from Georgia since 2000 — both political parties remained on edge over the unresolved Ossoff-Perdue race and its implications for the next two years in American politics. Whichever party wins that race will control the Senate, with Republicans counting on Mr. Perdue to prevail and give them the ability to constrain Mr. Biden’s policy ambitions.After Mr. Biden’s triumph in November, Mr. Warnock’s victory provides yet another comeuppance for the Trumpist politics that have come to define the Republican Party over the past four years. Ms. Loeffler had rebranded herself as a hard-line Trump loyalist to fend off a challenge from the right in the first round of voting. In recent weeks, she has continued to embrace the president, even using an election-eve rally with Mr. Trump in Northwest Georgia to proudly declare that she will oppose certifying his loss to Mr. Biden when Congress meets on Wednesday. Mr. Warnock and Mr. Ossoff ran as a virtual package deal, as did the two Republicans, often appearing at events together and crafting similar messages about the stark consequences for the nation if the other side won.Republicans used much of the runoff to focus on Mr. Warnock’s sermons, a line of attack that appeared to mobilize African-American voters, especially in more conservative rural Georgia where the church is a pillar of many communities. Mr. Trump’s refusal to acknowledge his defeat also robbed Ms. Loeffler of what might have been her best argument in what is still a slightly right-leaning state — that she would be a check on the liberal excesses in a government fully controlled by Democrats. Even before polls closed on Tuesday, senior Republican campaign officials were pinning the blame on the president, noting that their polling testified to the power of the “check-and-balance” argument that the party was unable to make because of Mr. Trump’s denial of the election results.The election was a tumultuous coda to Mr. Trump’s presidency, with control of the Senate and the first two years of Mr. Biden’s term in the balance. The runoffs were also an important bellwether for a Deep South state where once-dominant Republicans have begun to see their advantage slip because of an increasingly diverse electorate and the changing preferences of suburban voters.Election-day turnout was pivotal for Republicans, who were playing catch-up to Democrats. During an early-voting period that ended last week, more than three million Georgians cast their ballots, and turnout was heavy among African-Americans and in liberal bastions around Atlanta.For voters, the choice between the two pairs of candidates was stark: Mr. Perdue, 71, and Ms. Loeffler, 50, are both white millionaires who leaned into more conservative policy positions like gun rights and opposition to abortion. They also made the case to voters that their business success gave them real-world experience in handling economic matters.Mr. Ossoff, 33, and Mr. Warnock, 51, were a more diverse team. Mr. Ossoff, who is Jewish, is the head of a video production company and worked as a congressional aide. Mr. Warnock is a prominent pastor at the church in Atlanta where the Rev. Dr. Martin Luther King Jr. once preached. Both men promised a more robust response to the coronavirus pandemic and an expansion of Medicaid under the Affordable Care Act, and they embraced the national Democratic Party.They also railed against Mr. Trump, who ensured that he was regularly on Georgia voters’ minds with his incessant — and groundless — insistence that he was robbed of victory in the state by a “rigged” general election in November.Soon after his narrow defeat, Mr. Trump spurred a Republican civil war in Georgia, lashing out at two fellow Republicans, Gov. Brian Kemp and Secretary of State Brad Raffensperger, when they refused to take steps to alter the presidential results. Mr. Perdue and Ms. Loeffler, both ardent defenders of Mr. Trump, chose sides early, accusing Mr. Raffensperger of incompetence and calling on him to resign a few days after the election.Right up to the eve of the runoff vote, Republicans worried about the potential fallout from Mr. Trump’s efforts to overturn his defeat, particularly the revelation that he had called Mr. Raffensperger on Saturday and pressured him to “find” the votes that would help the president declare victory. Mr. Trump’s spurious claims of fraud stoked fears among some in his own party that his supporters would take him literally and sit out the election on the grounds that their votes would not amount to much in a compromised system.Even on Tuesday afternoon, when his party was fervently pushing election-day turnout, Mr. Trump was calling into question the integrity of Georgia’s election system. He asserted via Twitter that voting machines “are not working in certain Republican strongholds.” Mr. Raffensperger said the issues were minor and resolved by 10 a.m.Well before Tuesday, the president left the two Republican senators with a tricky task: arguing that Republican control of the Senate was crucial to constraining Mr. Biden, without conceding that Mr. Biden had actually won the presidency, which would punch a hole in Mr. Trump’s false narrative.To the end, though, both candidates hitched themselves to Mr. Trump, calculating that the party’s rank and file would sit out the runoff if they distanced themselves in the slightest.If the Republican hopefuls contorted themselves to accommodate Mr. Trump’s die-hard supporters — and risked alienating Biden-backing suburbanites in the process — the two Democrats did little to defy their own party.In a state where Republicans hold every statewide office, Mr. Warnock and Mr. Ossoff ran as national Democrats rather than emphasizing any differences they had with party orthodoxy, in the fashion of an earlier generation of Georgia Democrats. While resisting some of the ambitions of the far left, like defunding the police, the two candidates expressed support for abortion rights and gun control.Republicans seized on these issues, as well as the biographies of Mr. Ossoff and particularly Mr. Warnock, as they argued that the two men were too liberal for Georgia.Mr. Ossoff was mostly known for having run and lost in an expensive, hotly contested special House election at the outset of the Trump era in 2017. Among Mr. Warnock’s main challenges, by contrast, was the length of his record as a public figure and an activist preacher.After Mr. Warnock largely escaped criticism in the November election, when Ms. Loeffler was focused on fending off a challenge from her right, he came in for particularly harsh criticism.Republicans spotlighted Mr. Warnock’s most controversial sermons and portrayed him as a critic of the military and law enforcement. Mr. Warnock sought to defuse the criticism and soften his image by airing tongue-in-cheek commercials featuring him with a puppy.Mr. Ossoff also took some hard shots at Mr. Perdue, calling him a “crook” over controversial stock trades the senator made, while accusing him of trying to profit off the coronavirus pandemic, something Mr. Perdue denies.Neither party lacked for resources to make its arguments. These were the most expensive Senate contests in U.S. history. Including the campaigning before the runoff, more than $469 million was spent in the Perdue-Ossoff contest, according to the Center for Responsive Politics, and more than $362 million was dedicated to the Loeffler-Warnock race.That the races were competitive at all was a testament to the changing nature of Georgia.Though dominated by Republicans for much of the past two decades, the state is shifting because of an influx of newcomers, immigrants and American-born voters, chasing warm weather and Sun Belt opportunity. Democratic hopes were buoyed not only by Mr. Biden’s victory, but by the 2018 campaign of Stacey Abrams, who ran a competitive but unsuccessful race for governor.And the two Senate aces were pushed into runoffs by some of the defining forces shaping national politics.Mr. Ossoff made his political debut in 2017 as a fresh-faced and virtually unknown candidate vying for an open House seat in suburban Atlanta. The special election served as one of the first major referendums on Mr. Trump; Mr. Ossoff, despite his obscurity, was inundated with money from energized liberals across the country.Mr. Ossoff lost the 2017 race, but he carried his experience and name recognition into the 2020 battle, where he forced Mr. Perdue into a runoff.In the other race, Ms. Loeffler was appointed to the Senate by Mr. Kemp in December 2019 to fill the seat of Senator Johnny Isakson, who retired because of health concerns. But his choice displeased Mr. Trump, who had wanted Mr. Kemp to tap Doug Collins, then a hard-right congressman from Georgia who had served as one of Mr. Trump’s most loyal defenders during his impeachment.Mr. Collins jumped into the race anyway, forcing Ms. Loeffler far to the right; at one point she aired an ad in which she said she was more conservative than Attila the Hun.The strategy helped Ms. Loeffler win a place in the runoff, but seemed to invalidate Mr. Kemp’s original rationale for appointing her as she rebranded herself as a hard-line Trump loyalist.Rick Rojas, Astead W. Herndon and Sean Keenan contributed reporting.