Fulton County, which includes most of Atlanta, is expected to finish counting its ballots overnight.

ATLANTA — In a nail-biting scenario whose resolution could help determine the winner of a tumultuous presidential race, the elections director in Fulton County said Wednesday that Fulton — the state’s most populous county, which encompasses most of Atlanta — is expected to finish counting absentee ballots overnight.

The counting in Fulton County, which is a Democratic stronghold, may not end until after midnight, the director, Richard L. Barron, said Wednesday. “As long as it takes, we are going be here,” he told reporters.

Across Georgia on Wednesday night, the margin between the candidates continued to narrow. About 33,000 votes separated the presidential candidates, with President Trump leading Joseph R. Biden Jr. by less than one percentage point, with about 49.7 percent of the vote.

The closely contested presidential race in Georgia, which awards 16 electoral votes, underscored the fact that this Deep South state, once a reliable Republican stronghold, has become a legitimate battleground.

The fact that ballots remained uncounted on Wednesday came as little surprise. Voters were allowed to deposit absentee ballots in county drop boxes until 7 p.m. Tuesday. The process of counting them is labor-intensive, involving manually removing ballots from envelopes and, in some cases, subjecting them to human review.

But there were also unwelcome surprises, most notably a pipe that burst Tuesday morning in State Farm Arena, the stadium where the Atlanta Hawks play basketball and where Fulton County was tabulating votes. The plumbing failure, announced by county officials late Tuesday night, delayed the counting of an estimated 50,000 ballots.

The Trump campaign, which has been challenging election results in several counties and states, announced a lawsuit Wednesday over 53 absentee ballots in Chatham County, Ga., that suggested they may have arrived too late to be counted.

In a news conference Wednesday morning, Georgia’s secretary of state, Brad Raffensperger, said that the largest number of outstanding ballots — which at that point numbered more than 60,000 — were from Fulton County. About 50,000 ballots had been from DeKalb County, a Democratic-leaning area that also includes part of Atlanta, and roughly 7,000 ballots were from Forsyth County, which voted heavily for Mr. Trump in 2016.

Mr. Raffensperger, a Republican and supporter of Mr. Trump, said he would pressure county officials in the state to complete their tallies on Wednesday.