Sabathia took the no-hitter into the eighth inning but Kelly Shoppach's liner to left with two outs broke up the veteran left-hander's bid to hurl the first Yankees no-no since David Cone's perfect game against Montreal on July 18, 1999.
For the game, Sabathia (1-0) allowed one hit and two walks and struck out five. He was pulled following Shoppach's hit after tossing 111 pitches, 69 for strikes.
Cano pounded a two-run homer, Brett Gardner chipped in with two hits and two RBI, and Mark Teixeira broke an 0-for-17 slump to begin the season with an RBI double in the fifth inning in New York's victory.
Wade Davis (0-1) took the loss for the Rays, allowing four runs on seven hits and four walks over six innings.