Cobb (3-1) gave up three hits, one of them when Brett Gardner singled with one away in the ninth. Fernando Rodney then came on to preserve the three-run lead, but immediately gave up a single to Ichiro Suzuki.
The Rays took two from the Yankees in the three-game series and won for the sixth time in seven games.
Andy Pettitte (3-1) gave up seven hits and three runs over six innings in his first setback of the season. He struck out a season-high 10.
Tampa Bay's recent run of success has come after the Rays won just four times in their opening 14 games.
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