Tampa Bay avoided a three-game series sweep with David Price becoming the first American League pitcher this season to reach eight wins.
Ben Zobrist put the Rays in front for good with an RBI single in the third for a 2-1 lead. B.J. Upton drove in Zobrist later in the inning with a sacrifice fly and Sutton's two-run double one inning later gave Tampa Bay a 5-1 advantage.
Price (8-3) threw 109 pitches over just five innings, but gave up only one run on three hits with eight strikeouts.
Five relievers combined to surrender two runs on three hits while striking out five.
New York starter CC Sabathia (7-3) took the loss, giving up five runs on seven hits in seven innings. He struck out a season-high 12.
Yankees first baseman Mark Teixeira was in the lineup as usual despite learning earlier in the day he had damaged vocal cords that may take as long as a year to heal. He was hitless in three at-bats with a walk and was struck by a pitch during the ninth inning.