Mr. Pence repeatedly disputed quotations from Mr. Trump that were, in fact, accurate: He claimed in one instance that Mr. Trump had not indicated he might allow more nations to acquire nuclear weapons, though Mr. Trump had done just that. At one point, Mr. Pence appeared to break openly with Mr. Trump in his views about Russia, calling Mr. Putin a “small and bullying leader,” though Mr. Trump had consistently said the opposite.

Still, Mr. Trump’s campaign treated the aftermath of the debate as a kind of respite from a turbulent week and a half during which Mr. Trump made a series of intemperate attacks on subjects including a former Miss Universe’s physique and Mrs. Clinton’s marriage.

Mr. Pence took something of a victory lap at a campaign stop in Harrisonburg, Va., where Ed Gillespie, a former chairman of the Republican National Committee, introduced him as “one heck of a debater.” Mr. Pence said Mr. Trump had called to congratulate him, and declared the night a victory for the top of the ticket.

“From where I sat, Donald Trump won the debate,” Mr. Pence said. “Donald Trump’s vision to make America great again won the debate.” He repeated his accusation from the debate that Mr. Kaine and Mrs. Clinton have been throwing “an avalanche of insults” in the campaign.

There is scant polling information on how most voters reacted to the debate, and it is unclear that an encounter between comparatively low-key running mates might reshape a race dominated by two outsize presidential candidates. And any lingering fallout from the candidates’ performances will almost certainly be swallowed up by a debate between Mr. Trump and Mrs. Clinton on Sunday — a far more consequential event, and one much likelier to shift voter attitudes.