Pinch-hitter Martin Prado connected for a two-run shot immediately after Cobb was pulled in the eighth, and the Yankees climbed out of a four-run hole for the second consecutive night to take two of three in the series.
Young homered off closer Jake McGee with one out to win it. Clinging to slim playoff hopes, New York finished 5-4 on its penultimate homestand.
Yunel Escobar homered twice and drove in four runs for Tampa Bay.
Young's one-out double in the eighth ended Cobb's bid to become the first visiting pitcher to throw a no-hitter at Yankee Stadium since Virgil Trucks did it for the Detroit Tigers in 1952. Six pitchers combined on a no-hitter for the Houston Astros in the Bronx on June 11, 2003.
Cobb was lifted after Young's opposite-field drive up the right-center alley on his 102nd pitch. The right-hander with the funky windup, on quite a roll lately, received a warm hand from a crowd of 32,627 that included former Yankees great Mariano Rivera.
Prado, on the bench because of a sore hamstring, homered off Brad Boxberger to trim Tampa Bay's lead to 4-2.
Moments later, Derek Jeter was hit by a pitch in the left elbow and bent over in obvious pain. Jeter stayed in the game, and X-rays were negative.
New York got a bigger scare in the ninth when Chase Headley was hit on the chin by a 98 mph fastball from McGee (4-2). Headley stayed down on his back for a couple of minutes as he was attended to by two trainers.
With blood trickling down his chin, Headley sat up and walked off the field with both trainers.
Ichiro Suzuki followed with a double and Young, cut Aug. 9 by the crosstown Mets after a season-long slump, drove an 0-1 pitch to left field for his latest big hit in pinstripes.
New York Yankees starting pitcher Michael Pineda delivers in the first inning of a baseball game aga …
Shawn Kelley (3-5) got two outs for the win.
Cobb was in complete control until he walked consecutive batters with one out in the seventh. But he quickly regrouped, striking out Headley on a check-swing and retiring Suzuki on a foul pop.
Before that, the only runner allowed by Cobb came on an error by center fielder Kevin Kiermaier, a defensive standout who dropped Stephen Drew's long fly to left-center leading off the third.
Cobb struck out four and hasn't yielded more than two runs in any of his past 12 starts, a club record and the longest active streak in the majors.
''I didn't realize it was that good,'' Rays manager Joe Maddon said before the game.
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Young's 3-run HR in 9th rallies Yankees over R …
Tampa Bay Rays starting pitcher Alex Cobb delivers in the first inning of a baseball game against th …
Escobar connected twice off starter Michael Pineda, including a three-run shot, for the first multihomer game of his career.
TRAINER'S ROOM
Rays: Maddon said there was nothing new to report on ailing CF Desmond Jennings (sore left knee), who might miss the rest of the season. Jennings sat out his 13th consecutive game.
Yankees: LF Brett Gardner missed his fifth straight game with an abdominal injury. New York manager Joe Girardi said he hopes Gardner might be able to play this weekend. ... DH-OF Carlos Beltran (right elbow) sat out for the second straight game.
Rays: RHP Nathan Karns looks for his first major league win when he makes his Tampa Bay debut Friday night in Toronto. Karns made three starts for Washington last year.
Yankees: RHP Brandon McCarthy (6-4, 2.79 ERA) and rookie RHP Bryan Mitchell will be the starters Friday when the Yankees begin a seven-game road trip with a day-night doubleheader at first-place Baltimore. Mitchell has pitched in one big league game, out of the bullpen on Aug. 10, but he was primarily a starter in the minors.
