Soriano started the winning rally with a one-out double off Jamey Wright (2-2). He stole third and came home on Granderson's fly ball to center.
"He ends up getting himself safe at third and now the situation changes," Granderson said. "I just have to get a ball to the outfield."
Yankees manager Joe Girardi gave Soriano the green light to steal.
"If it's no outs you can't do that," Girardi said. "But with one out, we're taking a chance trying to make it easier on Grandy."
Soriano stumbled on his way to third.
"Being the DH, my leg is not loose 100 percent," Soriano said. "My mind is 100 percent, but legs are not."
Mariano Rivera, making his last regular-season appearance at Tropicana Field, received a standing ovation before getting the final three outs for his 38th save. The closer is planning to retire after this season.
Rivera has 37 saves at Tropicana Field, the most by any opponent. Second on the list is Jonathan Papelbon with 10. Rivera has converted 64 of 66 save opportunities overall against Tampa Bay.
Boone Logan (4-2) got two outs for the win.
Pinch hitter Alex Rodriguez opened the 10th with a single to center off Wright and went to second on Brett Gardner's sacrifice. Rodriguez then was doubled off second after Wright grabbed a line drive by Ichiro Suzuki that appeared headed toward center field.
Rodriguez, who is appealing his 211-game suspension by Major League Baseball, was booed every time he came to the plate during the series. He went 2 for 9 with two strikeouts.
Girardi had the third baseman on the bench at the start of the series finale, due to a day game after a night game. Rodriguez didn't make his season debut until Aug. 5 after having offseason hip surgery.
Evan Longoria homered and drove in two runs for the Rays.
"We just can't permit the stolen base at third base," Rays manager Joe Maddon said. "That's really what caused the win for them."
New York got a homer and two RBIs from Robinson Cano.
Longoria and Cano went back and forth for much of the game before Granderson delivered his key sacrifice fly.
Cano then hit an opposite-field homer to left in the fourth and a tiebreaking RBI double against Alex Cobb in the sixth. He was easily thrown out attempting to stretch the hit into a triple.
Longoria responded with his 28th homer in the bottom half.
Nova allowed six hits and tied a career high with six walks in 6 2/3 innings.
Cobb gave up seven hits over 5 2/3 innings. It was the right-hander's third start -- the other two were victories -- after missing two months after being struck in the head by a liner hit by Kansas City's Eric Hosmer on June 15.
"Two victories and a chance to sweep, you want to step on their throat and finish them off," Cobb said.
The Yankees plan to activate SS Derek Jeter (strained right calf) from the disabled list for Monday's game at Toronto. ... Tampa Bay All-Star LHP Matt Moore (sore left elbow) is scheduled to pitch in a minor league game Thursday and could rejoin the rotation next week. .... ... The announced crowd was 34,078, which was just the Rays' third home sellout this season.