Burnett pitched seven innings, allowing one unearned run on two hits for his 10th win of the season. He struck out five and walked two to remain undefeated since June 20.
Every Yankee recorded a hit in the win, and Nick Swisher homered twice to lead the Bombers' attack. It was Swisher's home run in the at-bat following Robinson Cano's solo shot in the sixth that chased Rays starter James Shields from the game.
Shields surrendered nine hits and five earned runs over 5 1/3 innings, taking the loss to start a series the Rays came in needing to win.
Trailing the Yankees by 6 1/2 games in the AL East and the Red Sox by four in the wild card, the Rays entered the series searching to take at least two of three from the Yankees. Now looking at a 7 1/2-game deficit in the East and standing five games back in the wild card, the Rays are treading water in the postseason race.
Swisher finished the game 2-for-5 with two home runs, three RBIs and two runs scored.
Cano was 2-for-3 with a home run, two RBIs, two runs scored and two walks.