Raphael Warnock and Jon Ossoff defeated Republican incumbents Kelly Loeffler and David Perdue respectively.

Democrats will control the Senate, the House of Representatives and the White House for the first time since 2009.

An estimated four million Georgians turned out to vote in the run-off election.

The result is a severe blow for outgoing Republican President Donald Trump.

The Georgia election was rerun because none of the candidates in the November general election achieved the 50% needed for victory under state rules.

Mr Warnock, a Baptist pastor, becomes the first black senator for Georgia - a slavery state in the US Civil War - and only the 11th black member of the Senate in US history.

Mr Warnock 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.

With the result in Georgia, the Democrats and the Republicans will now each have 50 seats in the Senate, the upper house of Congress.

However, the incoming Democratic vice-president, Kamala Harris, will preside over the Senate once she takes office and will have the tie-breaking vote.

This gives the Biden administration a greater chance of achieving its agenda on issues such as healthcare and climate change.

The Senate also has the power to approve or reject Mr Biden's nominees for cabinet and judicial posts.

It appears Republican worries about the two run-off elections in Georgia were well-founded. Their voters did not show up at the polls in the kinds of numbers they were hoping. Meanwhile, 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 ran as a team, and appeared to complement each other's electoral coalitions. Mr Warnock energised black voters across the state. Mr Ossoff drew out suburban and educated voters around Atlanta.

Donald Trump is set to receive considerable blame for the Republican defeat. 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.

Instead, the two races remained close, as Mr Trump spent most of his time and energy disputing his electoral defeat and lobbing attacks at Georgia's Republican leaders.

It turns out that may not have been a wise electoral strategy - and it will cost Republicans control of the Senate.

Both races were very tight. Mr Warnock defeated Senator Kelly Loeffler by a projected 50.7% to 49.3%, figures from the Associated Press indicate.

Ms Loeffler, who was appointed to the Senate last year to fill a vacancy, is a Trump loyalist.

The other contest was even closer. Jon Ossoff edged ahead of Senator David Perdue, 70, taking 50.28% to 49.72%.

Final results are expected later.

President Trump continued to make unsubstantiated claims of electoral fraud, doubting the integrity of the Georgia vote.

On Sunday, a recording emerged of Mr Trump putting pressure on Georgia's top election official, Brad Raffensperger, a fellow Republican, to "find" enough votes to overturn Mr Biden's win in the state.

Raphael Warnock, 51, is a pastor at the Ebenezer Baptist Church in Atlanta - the church where civil rights leader Martin Luther King once preached.

Mr Warnock was a youth pastor in New York City in the 1990s before moving to Baltimore, one of the poorest cities in the US.

There, he worked to educate his congregation about the risks of HIV/Aids. He also supports access to abortion, making him a rarity among faith leaders, and opposes the death penalty. In 2013, he gave the benediction at the second inauguration of President Barack Obama.

Jon Ossoff, 33, is a documentary filmmaker, born in Atlanta. As a high school student, he interned for John Lewis, the civil rights campaigner and congressman who died in 2020.

He became an aide to a Democratic member of Congress, specialising in foreign policy and national security.

In 2012, Mr Ossoff joined an independent production company specialising in investigative journalism, Insight TWI, which is based in London. He is now the CEO.