Biden declared victor in hard-fought race for Pennsylvania.

PHILADELPHIA — Joseph R. Biden Jr. defeated President Trump in Pennsylvania, winning its 20 electoral votes and presidency.

Mr. Biden had steadily erased Mr. Trump’s early lead in the state — at one point, the president led by half a million votes — as ballots, mostly absentee and mail-in votes, were counted over the past few days. Most of the remaining uncounted votes in the state are in Democratic-leaning areas.

The Biden campaign hoped further counting could push its lead above 0.5 percent, obviating the need for a recount there and setting the stage for victory. [After this post was published, Mr. Biden’s lead in Pennsylvania reached above 0.5 percent and news outlets declared him president-elect.]

The biggest fight in the state has been over ballots that were postmarked by Election Day but arrive later. Nearly a dozen lawsuits filed by Mr. Trump and his allies are working their way through the courts in Nevada, Pennsylvania, Michigan and Georgia, trying — so far unsuccessfully — to stop ballot counting and invalidate enough votes to erase Mr. Biden’s leads there. In September, the state Supreme Court ruled, over Republican objections, that election officials could accept ballots arriving up to three days later. The U.S. Supreme Court declined to intercede, but left open the possibility that it could revisit the question.

Separately, the Supreme Court did grant the Trump camp a minor victory in Pennsylvania on Friday evening, when Justice Samuel A. Alito Jr. ordered election officials there to keep the late-arriving ballots separate from other ballots, and not to include them, for now, in announced vote totals. But the victory was essentially in name only: Pennsylvania’s secretary of state had already given that instruction.

In Allegheny County, a predominantly Democratic area that includes Pittsburgh, election workers were going through roughly 20,000 mail-in ballots and additional provisional ballots on Saturday, Rich Fitzgerald, the county executive, said in a televised interview.

The county’s mail-in ballots have so far been won overwhelmingly by Mr. Biden, as have the provisional ballots.

Mr. Fitzgerald cautioned that the last ballots to count would be the trickiest, requiring additional checks to ensure they were not duplicates, which could slow the process.

Responding to baseless allegations by the Trump campaign of vote-counting secrecy, he said that observers and journalists had access to the vote-counting site and that there were as many surveillance cameras there as in a casino.