Pastor Raphael Warnock is projected to win one seat. Fellow Democrat Jon Ossoff leads narrowly in the other.
If they both win, Mr Biden will control Congress fully and have a much better chance of pushing through his agenda.
He said it was "time to turn the page. The American people demand action and they want unity".
Why is there an election in Georgia?
The election is being rerun because of Georgia's rule that a candidate must take 50% of the vote in order to win.
None of the candidates in November's general election met that threshold.
With 98% of votes counted, US TV networks and the Associated Press news agency called the first of the two races for Mr Warnock.
Control of the Senate in the first two years of Mr Biden's term will be determined by the outcome of the second run-off.
Mr Warnock is set to become the first black senator for the state of Georgia - a slavery state in the US Civil War - and only the 11th black senator in US history.
He serves as the reverend of the Atlanta church where assassinated civil rights leader Martin Luther King Jr grew up and preached.
Claiming victory, he paid tribute to his mother, Verlene, who as a teenager worked as a farm labourer.
"The other day - 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," he said.
If both Democrats win, the Senate will be evenly split 50-50, allowing incoming Democratic Vice-President Kamala Harris the tie-breaking vote. The Democrats narrowly control the House of Representatives.
Mr Ossoff has also claimed victory in his race against Republican Senator David Perdue, but that race is even tighter. At 33, he would be the Senate's youngest member for 40 years.
A Democrat has not won a Senate race in Georgia in 20 years but the party has been boosted by Mr Biden's presidential election win over Mr Trump there. Mr Biden's margin of victory was about 12,000 votes among five million cast.
Mr Biden is due to be inaugurated as president on 20 January. He won 306 votes to Mr Trump's 232 in the US electoral college, which confirms the president.
Mr Biden won at least seven million more votes than the president.
When will we get a result?
Final results are expected later on Wednesday.
The margins are extremely tight. Mr Warnock is projected to have won his race by 50.6% to 49.4% over incumbent Senator Kelly Loeffler.
Edison Research, which supplies election results to news organisations including the BBC, says Mr Ossoff is leading by 50.2%, to 49.8% over Mr Perdue.
More than 98% of ballots from Georgia's 159 counties have now been counted. The remaining votes come from the Atlanta suburbs, which are projected to go heavily for the Democrats.
More than three million voters - about 40% of those registered in Georgia - voted before Tuesday. Early voting benefited Mr Biden in November's White House election.
Trump will face blame if Republicans lose
Although the results are not final, it appears Republican worries about the two run-off elections in Georgia were well-founded. Their voters did not show up in the kinds of numbers they were hoping. Democrats turned out at higher levels. In county after county, both Jon Ossoff and Raphael Warnock outperformed their general election numbers.
The two Democrats, at times running as a team, appeared to complement each other's electoral coalitions. Mr Warnock energised black voters across the state. Mr Ossoff, although he slightly underperformed his Democratic counterpart, still attracted suburban and educated voters around Atlanta.
If it turns out both prevail, Donald Trump will receive considerable blame for the Republican defeats. The party that loses the White House usually does better in subsequent congressional elections, not worse. And Georgia, despite Joe Biden's victory there, is still a traditionally conservative state.
But Mr Trump spent most of his time and energy disputing his electoral defeat and lobbing attacks at Republican leaders in the state.
It turns out that may not have been a wise electoral strategy - and it could cost Republicans control of the Senate.
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January 07, 2021 at 03:04AM
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