Damon had a two-run double, and Derek Jeter came up with a clutch home run to back Clippard, a 22-year-old right-hander who was summoned from Triple-A Scranton/Wilkes-Barre and added to the Yankees' roster before the game.
Jorge Posada and Alex Rodriguez also homered for the Yankees, who logged nine extra-base hits in a game for the first time this season and evaded a series sweep at the hands of their crosstown rivals.
"I thought the ballclub was very loose tonight," said Yankees manager Joe Torre, before he amended his statement. "Loose, considering where we are and what we've done lately. I think everyone played off the kid tonight."
In the midst of one of the most challenging stretches of their schedule, the path doesn't get much easier for the Yankees, who welcome the American League East-leading Red Sox to town on Monday.
But their spirits rose markedly after a surprising and impressive performance by Clippard, who filled a hole left in the rotation after heavy rains in Chicago forced a doubleheader last week and bumped Mike Mussina from his expected spot.
Clippard took advantage of the opportunity, limiting the potent Mets lineup to three hits and striking out six in a 95-pitch performance, his only blemish being a second-inning home run by David Wright.
He said that the state of the Yankees' struggles were no secret around the organization's Triple-A affiliate, certainly not with what seems to be a constant shuffle of pitchers checking in and out of both clubhouses.
"I don't think it makes it any tougher," Clippard said. "It'd be tough if we were 10 games up or 10 games behind. You're making your Major League debut and you're going to be nervous and things are going to go through your mind. I was lucky enough to succeed tonight."
Buoyed by Clippard's first turn on the big stage, the Yankees broke out with a strong showing against Mets right-hander John Maine, who was bounced after five innings.
After leaving five runners on through the first three innings, the Yankees cracked through with four runs in the fourth inning.
Maine offered walks to Bobby Abreu and Doug Mientkiewicz before Damon came through with a bloop double to right-center field in front of diving right fielder Shawn Green, scoring the Yankees' first two runs.
Jeter followed with his third home run of the season, a two-run shot to left and his second in as many games -- a timely shot that Torre said reminded him of Jeter's home run in this very ballpark, sparking the Yankees in Game 4 of the 2000 World Series as he teed off on the first pitch by Mets starter Bobby Jones.
Told of Torre's flashback to a Fall Classic performance seven years removed, Jeter's first inclination was to laugh and say, "Really?"
"I don't hit very many home runs," Jeter said. "I hit a few, but you're not trying to -- especially in that situation. Johnny's on second with two outs, and you're just trying to get a hit. Two-out hits with runners in scoring position, they're tough to get. I was fortunate it went out."
Maybe Torre's comparison wasn't perfect, but the point was valid. For days, the Yankees have wondered longingly when their high-octane offense would begin to pump, and Jeter's deep drive seemed to confirm that more prosperous days could be had.
"We needed it," Jeter said. "We really haven't been scoring runs in bunches -- one run here, one run there. Any time you can go out and get three or four runs, it's good, especially with Tyler throwing the way he was. You want him to get comfortable."
In the fifth inning, Posada also homered for the second straight game, reaching Maine, who allowed five runs and eight hits, for a solo blast to right and extending his hitting streak to a career-high 15 games.
Rodriguez clubbed his Major League-leading 17th home run off right-hander Ambiorix Burgos in the seventh to provide the necessary insurance for Clippard, who became the seventh rookie pitcher to make a start for the Yankees this season.
Before leaving his Scranton/Wilkes-Barre teammates last week, ticketed to fly north to New York from his team's series in Norfolk, Va., Clippard half-jokingly vowed that he'd "hit a bomb" under the National League-style rules at Shea Stadium.
With only limited batting practice to his credit, going through soft-toss drills with the Triple-A coaching staff after having been informed of his pending promotion, Clippard proved to be no slouch with the bat.
He dropped down a successful sacrifice bunt in the third inning and even added his first Major League hit in the victory, stroking a one-out double to right-center field off Mets reliever Scott Schoeneweis in the sixth.
That baseball was retrieved by a batboy and tucked away for safe-keeping. As he addressed reporters in an empty visitors' locker room following the game, Clippard also pointed out a game ball -- the final out thrown by Mariano Rivera, who struck out Julio Franco -- and an assortment of lineup cards provided by the umpiring crew.
For Clippard, who left his New Jersey hotel room around 2:15 p.m. ET so as not to be late for his big date, it had been a long, eventful day. As he assembled his mementos and prepared for whatever the future might hold after this, it was certainly worth the wait.
"I was hoping to come in here and at least give them a chance," Clippard said. "To pitch like I did is unbelievable. This is the best day of my life, for sure. I'm just glad I could get us a win."