Days of pent-up post-election anxiety gave way to spontaneous street celebrations as Joe Biden supporters on Saturday cheered news of his presidential win with a cacophony of banging pots and pans, honking horns and dance music in cities across the United States.

Rallies by hundreds of supporters of Republican President Donald Trump, who insisted on Saturday “this election is far from over,” were more muted in size and scope compared with the festive crowds that turned out by the thousands in some cities for Biden...