WASHINGTON — As the Senate voted Wednesday night on a bipartisan deal to reopen the government and avert a national default, President Obama emerged from the Oval Office prepared to head to the White House briefing room to deliver a televised statement. But he was thinking beyond the moment.

Pausing near his secretary’s desk, Mr. Obama reached into his in-box and found the draft remarks prepared for the next day in which he planned to move past the confrontation. He sat down in a chair and began editing the speech. This was the moment to push ahead, he told aides, to press his advantage and lay out specific areas where he could make progress with Republicans.

By nearly all accounts, Mr. Obama emerged the winner of the showdown, having stared down attempts to undercut his health care program or force other concessions, but it is not clear what he actually won. Did he change the dynamic of his tumultuous presidency and break the cycle of Washington gridlock, opening the way to more meaningful legislation in months to come? Or did he merely kick the can down the road three months so he and Congress will be in the same place again, repeating a pattern that will define his remaining three years in office?

The president and his team hope that Congressional Republicans, stung by this week’s defeat, re-evaluate their scorched-earth strategy and seek agreement with Mr. Obama on issues like immigration and long-term spending to demonstrate that they can govern. But they fear the opposition may instead double down and become even more determined to prevent the White House from getting the upper hand in future legislative battles.