“We’re going to win this race,” Biden says as his lead grows in Pennsylvania and Georgia. – The New York Times

WILMINGTON, Del. — Joseph R. Biden Jr. stood on the cusp of the presidency on Saturday, seizing leads over President Trump in Pennsylvania and Georgia and building on his lead in Nevada as he drew ever closer to securing the 270 electoral votes needed to lay claim to the White House.

“We’re going to win this race with a clear majority of the nation behind us,” Mr. Biden said late Friday night as he recounted his vote totals in the major battleground states and projected confidence that victory was inevitable. “We’ve rebuilt the ‘blue wall’ in the middle of the country that crumbled just four years ago. Pennsylvania, Michigan, Wisconsin, the heartland of this nation.”

Mr. Biden, who was winning the popular vote by more than four million votes on Friday night and has already won 253 electoral votes, had pulled ahead of Mr. Trump in Pennsylvania by more than 28,800 votes. If his lead holds — and it is expected to — the state’s 20 electoral votes would vault him past the threshold to win the election.

In his remarks at an event center in Wilmington, Del., Mr. Biden claimed a “mandate for action” to begin work on issues that included the coronavirus, the economy, climate change and systemic racism.

But he also urged patience and calm as the results came in.

“We have to remain calm, patient, let the process work out as we count all the votes,” he said.

In Philadelphia, some Biden supporters started dancing in the street outside the convention center, in front of a banner declaring “The People Have Spoken.”

In Georgia, Mr. Biden’s lead was initially so narrow that state officials said a recount was inevitable. But by Friday evening it had grown to about 4,000 votes.

In Arizona, Mr. Biden maintained his lead as election officials continued to plow through tens of thousands of ballots from Phoenix and its suburbs. His advantage shrank slightly, but not by as much as Republicans had hoped.

In Nevada, Mr. Biden nearly doubled his lead Friday to around 22,600 votes.

Mr. Biden’s appeal to let the process play out contrasted with that of Mr. Trump, who took the lectern in the White House briefing room on Thursday to falsely claim that the election was riddled with fraud, as part of an elaborate coast-to-coast conspiracy by Democrats, the news media and Silicon Valley to deny him a second term.

As the number of outstanding ballots slowly dwindled, Mr. Trump was left increasingly with only legal challenges to forestall defeat, while Mr. Biden was betting on the steady accumulation of mail-in ballots to keep him on top in Pennsylvania.

Mr. Biden had already begun to project the image of a man preparing to assume the mantle of the presidency, meeting with his economic and health advisers to be briefed on the coronavirus pandemic.

Mr. Biden kept a low profile for much of Friday. But outside the Westin hotel near his home in Wilmington, signs of celebration were afoot — a celebration that would end up being delayed, again, as Mr. Biden waited on an official call of the race.

Thomas Kunish, 40, of Mechanicsburg, Pa., 100 miles away, said he had driven in to show support with his five-year-old son, spending the night in the car. The last time he voted, he said, was for George W. Bush in 2000.

“It was interesting, the past four years,” Mr. Kunish, who works in the defense industry, said of Mr. Trump’s presidency. “There was hope when he got elected, things maybe would change?” Instead, he said, the Trump administration was marked by “turmoil.” He and his son were hoping to see fireworks Friday night — which did not come to pass.

Mr. Biden’s team had hoped the former vice president would have delivered a victory speech in prime time, but instead, Mr. Biden spoke briefly before a small group of reporters late Friday night.

“I hope to be talking to you tomorrow,” Mr. Biden said.