Sabathia got the best of Tampa Bay ace David Price, enabling the Yankees to win two out of three in the series.
Tampa Bay managed an unearned run in both the first and second innings against Sabathia (5-0), but had only two hits over the next six frames. Sabathia threw a season-high 119 pitches.
Granderson hit a solo homer in the second to get the Yankees within one and they tied it later in the inning on an RBI single by Chris Stewart.
Robinson Cano put New York in front with a two-run homer during a three-run fifth.
Price (5-2) allowed five runs on 11 hits with four strikeouts in seven innings. It was Tampa Bay's fourth loss in five games.
Rafael Soriano gave up a hit and a run in the ninth, but held on to collect his first save. Soriano's performance came one night after David Robinson had a blown a save for the Yankees, who are seeking a reliable closer for the rest of the season in the absence of the injured Mariano Rivera.