Appearing before supporters in his hometown of Wilmington, Delaware, after midnight, Biden said "we believe we're on track to win this election."
"Keep the faith, guys, we're going to win this."
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But Biden, 77, warned that because of unprecedented use of mail-in voting during the coronavirus pandemic getting final results would "take a while."
Seconds later, Trump tweeted his rebuttal, saying: "We are up BIG, but they are trying to STEAL the Election."
The Republican president, who was watching results come in with staff at the White House, said he would shortly deliver a public statement.
"A big WIN!" he tweeted.
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The dramatic clash, which stoked fears of electoral chaos, came after early results showed the two candidates largely retaining states already in their column, but without delivering the kind of knock-out blow that could decide the final tally quickly.
Results confusion
Adding to the tension was confusion over the reporting of key results in the U.S. media.
All night after polls closed Tuesday, the two men traded state for state -- Trump retaining the giant prizes of Florida and Texas, while Biden kept Virginia and easily won New Hampshire, where Hillary Clinton only just eked out victory four years ago in her eventual loss to Trump.
But as both sides held on to states they were expected to win, the number of remaining states yet to declare dwindled, pointing to an ever-tighter end game and growing potential for disputes to end up in the courts.
At times, U.S. media organizations that analyze the data and pronounce winners added to the sense of uncertainty.
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There was huge buzz over Fox News' announcement that Biden had won previously Trump-held Arizona.
If confirmed it would be a real shift in the overall race's momentum. However, no other U.S. media made the Arizona call and the Republican governor of the state, Doug Ducey, said "it's far too early" to announce a result.
Hours passed before media groups were able to agree on calling Florida for Trump. And the results in crucial North Carolina and Georgia remained unclear into Wednesday, even with 95 percent and 85 percent of votes counted respectively.
No landslide
What seemed likely was that Americans would at least not see a landslide that Democrats had dreamed they might pull off if they could open with a win in Florida.
Instead, analysts said the tit-for-tat victories in states across the huge country could finally whittle down a final fight over just a handful of swing states -- especially Pennsylvania.
However, Pennsylvania and Georgia were among states where vote counting was going more slowly, complicated this year by the huge number of mail-in ballots. In many states, ballots that were postmarked by the election but being delivered by the postal service after are still legal, making it likely that poll workers will be still tabulating results for several days.
Trump has long targeted such votes, which are more likely to come from Democrats. He falsely claims that they are illegal and that only votes counted on Election Day itself can be valid.
"Votes cannot be cast after Poles are closed!" he tweeted Tuesday, wrongly spelling "polls," and wrongly characterizing the post-election arrival of such ballots.
Twitter flagged this tweet for violating its rules on misleading the public.
Race for Congress
Networks projected the Democrats to have maintained control of the House of Representatives, as widely expected, but their hopes of winning back the Senate met headwinds.
The Democrats flipped one Senate seat from the Republicans in Colorado, with former governor John Hickenlooper projected to triumph, but were also expected to lose an especially vulnerable senator in Alabama.
One notable win in the Senate was for the Republican leader, Mitch McConnell, who easily fended off a challenge in Kentucky.
And in Georgia, Republican Marjorie Taylor Greene -- a political newcomer who has promoted the far-right QAnon conspiracy theory -- won a seat in the U.S. House of Representatives, giving the widely debunked movement a voice in Congress.
Outside the White House, a boisterous, peaceful protest in a plaza renamed for the Black Lives Matter movement turned heated as the night wore on, with scuffling after a person appeared to throw a gas cannister.
In Portland, the center of confrontations this summer between leftist protesters and police, some 400 people marched toward the downtown under a watchful eye of state police.