Democrat Raphael Warnock has won one of Georgia’s two US Senate run-offs, becoming the first black senator in his state’s history and putting the Senate majority within the Democrats’ reach.

A pastor who spent the past 15 years leading the Atlanta church where Martin Luther King preached, Reverend Warnock defeated Republican incumbent Kelly Loeffler.

It was a stinging rebuke to outgoing President Donald Trump, who made one of his final trips in office to Georgia to rally his loyal base behind Senator Loeffler and the Republican running for the other seat, David Perdue.

The focus now shifts to the second race between Senator Perdue and Democrat Jon Ossoff.

That contest remains too early to call, with votes still being counted.

If Mr Ossoff wins, Democrats will have complete control of Congress, strengthening President-elect Joe Biden’s standing as he prepares to take office on January 20.

Reverend Warnock’s victory is a symbol of a striking shift in Georgia’s politics as the swelling number of diverse, college-educated voters flexes their power in the heart of the Deep South.

It marks the end of nearly two decades in which Democrats have been shut out of state-wide office and follows Mr Biden’s victory in November, when he became the first Democratic presidential candidate to carry the state since 1992.

Reverend Warnock, 51, acknowledged his improbable victory in a message to supporters early on Wednesday, citing his family’s experience with poverty.

“The other day, because this is America, the 82-year-old hands that used to pick somebody else’s cotton picked her youngest son to be a United States senator,” he said.

“Tonight, we proved with hope, hard work and the people by our side, anything is possible.”

Senator Loeffler refused to concede in a brief message to supporters shortly after midnight.

“We’ve got some work to do here. This is a game of inches. We’re going to win this election,” said Senator Loeffler, a 50-year-old former businesswoman who was appointed to the Senate less than a year ago by the state’s governor.

Senator Loeffler, who remains a Georgia senator until the results of Tuesday’s election are finalised, said she would return to Washington on Wednesday morning to join a small group of senators planning to challenge Congress’ vote to certify Mr Biden’s victory.

“We are going to keep fighting for you,” she said.

“This is about protecting the American dream.”

Georgia’s other run-off election pitted Senator Perdue, a 71-year-old former business executive who held his Senate seat until his term expired on Sunday, against Mr Ossoff, a former congressional aide and journalist.

At just 33 years old, Mr Ossoff would become the Senate’s youngest member if he secures victory.

Mr Trump’s false claims of voter fraud cast a dark shadow over the run-off elections, which were held only because no candidate hit the 50 per cent threshold in the general election.

He attacked the state’s election chief on the eve of the election and raised the prospect that some votes might not be counted even as votes were being cast on Tuesday afternoon.

Republican state officials on the ground reported no significant problems.