Acknowledging the tortured road he and other skeptical Democrats traveled, Mr. Schumer said, “I also have a great deal of respect for the careful thought and deliberation my colleagues went through,” adding, “I recognize for them that this is a vote of conscience just as it is for me.”

Mr. Obama’s triumph in securing the deal — without the support of a single member of the party now in control of Congress — is refashioning the definition of victory for a waning presidency in the era of divided government.

While bipartisan victories tend to be those most celebrated outside of Washington, success by the president is now often measured more by the scope of the policy achieved than by any claim of sweeping consensus. And losing has its own evolving meaning as well. Republicans will use Mr. Obama’s triumph — as they did with the health care law — as a means to attack Democrats in anticipation of next year’s election.

Indeed, Senator Mitch McConnell, Republican of Kentucky and the majority leader, said Thursday that he would force the exact same vote again next week, just to make Democrats go through the exercise one last time. “If the president’s so proud of this deal, he shouldn’t be afraid,” Mr. McConnell said, as he stared at Democrats on the floor immediately after the vote.