Republicans across the country yesterday hailed President Reagan's performance in the Sunday night television debate as "the final victory" in his reelection campaign, while Democrats sought to salvage a degree of optimism from what many acknowledged was an outcome that fell short of their hopes.

Former Democratic National Committee chairman Robert S. Strauss said Walter F. Mondale "crossed a threshold of credibility" with the substance of his responses, but he said the improvement in Reagan's performance since his faltering effort in Louisville two weeks ago raised a "serious problem" of Mondale "running out of time" to overhaul the incumbent.

As Reagan headed west, Missouri Gov. Christopher S. Bond (R) capsulized the GOP reaction by saying, "Any question that had been raised about the president's leadership and control of issues has been well settled by the debate."

"Reagan did what he had to do to win reelection," echoed Rick Robb, the co-chairman of the president's campaign in Pennsylvania.

On the Democratic side, New York Gov. Mario M. Cuomo (D), a leading cheerleader for Mondale after the first debate, held out the hope that "the reaction will be better in two or three days . . . as people focus on the substance . . . . I think we're still in the game."

But pollster Bill Hamilton, who has Democratic congressional clients, said he thinks that Reagan's margin "will close some, but I don't think the rate of change will be dramatic enough. The election is not on Thanksgiving Day."

Overnight voter surveys taken for Newsweek magazine, CBS News and ABC News showed the debate a virtual tie -- the result Mondale pollster Peter D. Hart said he had found. But a survey for USA Today called Reagan the winner.

Newsweek's poll gave Reagan the edge by 43 percent to 40 percent. ABC showed Reagan ahead, 39 to 36; CBS put Reagan the winner, 41 to 38. USA Today had it 44 to 27 for the president.

A number of Democrats commented that Mondale appeared less aggressive and dominating than he had in Louisville and that his personal appearance was disturbing.

"For whatever reason," said California Assembly Speaker Willie Brown (D), "he was not as direct, not as specific, not as strong on the issues as he had been the first time. He was equal to the president, but that is not good enough to create the movement we need."

Brown said Mondale looked "physically exhausted," and Democratic campaign consultant David Garth said, "The bags under his eyes looked unbelievable."

An adviser to Mondale blamed the problem on lighting. "It was not fatigue," said Frank Greer, a media consultant, suggesting that the impression may have resulted from the lighting level being "too low at the beginning" of the debate.

Whatever the cause, Republicans rejoiced that Reagan looked younger and appeared sharper than he had in Louisville.

White House chief of staff James A. Baker III, normally cautious, told reporters that Mondale would have to shift "a million votes a day" in the final two weeks of campaigning to overcome Reagan's lead.

Democrats scoffed at that estimate and said they saw some benefits from the final debate. Pollster Dotty Lynch said she believed that the issue of nuclear arms in space, on which Mondale repeatedly challenged Reagan, would be effective in converting the three out of 10 former backers of Sen. Gary Hart (D-Colo.) she said her surveys indicate support Reagan.

Garth suggested that Mondale, by emphasizing his readiness to be tough with the Soviets, might gain some support among blue-collar men. "He came across as Max Macho," Garth said.

Still, the New York consultant called a Mondale comeback "possible but not probable," and Republican pollster V. Lance Tarrance said his guess was that Reagan had prevented any major Mondale gains by "spending the evening reassuring those 'soft' voters that he has a balanced view of Soviet-American relations and a real concern for life on this planet in the next hundred years."