Appealing directly to Mr. Sanders’s supporters, he underscored the pivot Mr. Biden has been trying to make since wrapping up the nomination: from an argument, essentially, for restoring the pre-Trump status quo to an argument that this is insufficient. It is the argument Mr. Sanders and other progressive candidates — like Senator Elizabeth Warren of Massachusetts, whose call for “big structural change” Mr. Obama overtly echoed — made all along.

“To meet the moment, the Democratic Party will have to be bold,” Mr. Obama said. “I could not be prouder of the incredible progress that we made together during my presidency. But if I were running today, I wouldn’t run the same race or have the same platform as I did in 2008. The world is different. There’s too much unfinished business for us to just look backwards. We have to look to the future.”

Mr. Sanders’s ideas and his supporters’ enthusiasm would be critical in November, Mr. Obama said, before discussing several of the issues that drove Mr. Sanders’s campaign. Americans, he said, needed student debt relief that did “more than just tinker around the edges,” health care access that went beyond the Affordable Care Act, climate policies bolder than the Paris Agreement, and policies to address “the vast inequalities created by the new economy” — inequalities that he acknowledged had been evident long before now.

“Of course Democrats may not always agree on every detail of the best way to bring about each and every one of these changes, but we do agree that they’re needed,” he said. “And that only happens if we win this election.”

At points in his video announcement, which ran more than 12 minutes, Mr. Obama seemed to be doing more than endorsing his former vice president, more even than trying to unite his party. From his first words — “these aren’t normal times” — it was something like an Oval Office address to a battered nation, designed for maximum contrast with the office’s current occupant.