For the senators who are eyeing the White House, the consequences of a wrong vote on war are not abstract. Hillary Rodham Clinton’s vote for the invasion of Iraq when she was a senator opened a lane for the 2008 candidacy of Barack Obama. When Mr. Obama was in the Senate, his antiwar views helped him shape a distinctive public persona. John Kerry’s 2004 campaign for president was crippled by the accusation that he waffled on his vote for the Iraq invasion with his later criticism of it.

In their speeches on Thursday, senators like Mr. Paul and Marco Rubio of Florida, who voted for the measure and is another possible 2016 contender, were looking far beyond the vote. “Amid the interventionists’ disjointed and frankly incoherent rhetoric,” said Mr. Paul, “the only consistent theme is war. These barnacled enablers have never met a war they didn’t like.”

Mr. Rubio, a fiscal hawk who has always voted against short-term spending bills because he says they are the wrong way to fund the government, found himself squeezed between that principle and his position as the leader of the 2016 interventionist wing. “We are asked to decide things in this chamber that are in the best interest of our country,” he said, “even if they did not work out the way we wanted them to.” Democrats touted as possible presidential candidates who voted no included Senator Elizabeth Warren of Massachusetts and Senator Kirsten E. Gillibrand of New York.

The issue of military intervention in Syria is likely to resurface shortly after the midterms. Members of both parties are calling for a vote then on a use-of-force resolution that would have far broader implications than the one approved on Thursday.

Many liberal Democrats and conservative Republicans agree that the administration’s justification for using force today — the congressional authorization for using force granted after the Sept. 11, 2001, attacks — is specious. “We are living on borrowed time and we are traveling on vapors,” said Senator Richard J. Durbin of Illinois, the No. 2-ranking Democrat, arguing on Thursday that the old authorization had long expired.