Willy Aybar and Ben Zobrist homered to help the AL champion Rays overcome four errors, which doubled their previous season high.
Three were throwing errors by All-Star catcher Dioner Navarro.
The Rays have won four straight and six of seven. They are 3-0 at the new Yankee Stadium, including an 8-6 victory May 7 when Rivera served up back-to-back homers for the first time in his career. Carl Crawford and Evan Longoria connected for ninth-inning solo shots in that one.
Mark Teixeira homered in the eighth and hit a two-run double in the ninth for New York. Alex Rodriguez also went deep for the Yankees, who had won 17 of 22.
Rays reliever Dan Wheeler gave up three straight hits to start the ninth, then retired Rodriguez and Jorge Posada with Teixeira on second.
Ex-Yankee Randy Choate got Robinson Cano on a long fly to center for his third save in eight days - the first three of his major league career, which began in 2000. B.J. Upton made a lunging grab to end it.
Zobrist opened the Tampa Bay ninth with a triple to left-center and scored on a single by Dillon, a journeyman who was acquired from Oakland on May 9. He has three hits in each of the past two games, matching his career high each time.
Upton added a two-out RBI single off Rivera (0-2), who was then lifted from the game in a rare sight. Crawford hit a run-scoring single off Phil Coke, and another run scored on an error by Rodriguez at third.
Friday night's rainout pushed back the matchup between Sabathia and Price, drafted No. 1 overall by Tampa Bay in 2007.
The 23-year-old lefty walked five in 5 2-3 innings but allowed only two hits and three runs - one earned.
Sabathia went eight innings, giving up five runs (four earned) and five hits. He entered 4-0 with a 2.08 ERA in his previous five outings, and 7-1 with a 2.44 ERA in 11 career starts against Tampa Bay.
Tampa Bay's J.P. Howell (1-2) struck out pinch-hitter Hideki Matsui with runners at the corners to end the eighth.
The Rays took a 5-3 lead on Aybar's three-run shot in the sixth.
Aybar started at third base for Longoria, out since leaving Tuesday's game with tightness in his left hamstring. Longoria was intentionally walked as a pinch hitter in the ninth and removed for a pinch runner. He's expected back in the lineup Sunday.
Aybar dropped a popup just inside the third-base line for an error in the seventh.
Notes: Longoria ran in the outfield to test his leg before the game. ... Rays manager Joe Maddon said injured DH Pat Burrell (neck strain) likely will begin a minor league rehab assignment in the next few days and could be back in Tampa Bay's lineup by Friday.
... Navarro entered with one error all season. Since 1900, the major league record for errors in a game by a catcher is four, done four times - most recently in 1927 by Bill Moore of the Boston Red Sox. ... Rodriguez's 561st homer left him two behind Reggie Jackson for 11th on the career list. ... There has been at least one homer in all 27 games at the new Yankee Stadium.
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