Granderson hit a two-run homer in Detroit's three-run eighth inning and the Tigers beat the Indians 9-7 on Saturday.
Detroit blew a 5-0 lead and trailed 7-6 before its big inning against Rafael Betancourt (0-1), rallying to end an eight-game losing streak to Cleveland.
"That could have been a disaster," said Tigers manager Jim Leyland, whose team had lost three straight overall. "In the end, it turned out pretty well."
"We came in with a three-game skid, and it looked like we might make it four," Granderson said. "Now we've got a chance to win this series, and that's going to be big in this division."
Betancourt got the final two outs of the seventh, but couldn't get through the next inning.
"That's been a struggle for us," manager Eric Wedge said. "I still have a lot of confidence in Rafael Betancourt, and I expect him to do the job the next time, but we need someone else to step up and help do the job in the seventh and eighth innings to get the ball to Kerry [Wood]."
Joel Zumaya (1-0) picked up the win with 1 2-3 shutout innings of relief, and Fernando Rodney finished for his fifth save. Cleveland had runners on the corners with one out in the ninth, but Rodney struck out Mark DeRosa and Jhonny Peralta to end the game.
"Our bullpen is finally healthy, and we've got some great arms out there," Zumaya said. "That's going to help us win games like this."
Adam Everett hit his second career grand slam in the fourth for the Tigers.
Both starters struggled but ended up with no-decisions. Cleveland's Aaron Laffey allowed five runs -- as many as in his previous three starts combined -- and six hits with five walks in just 3 1-3 innings.
"Aaron was just inconsistent today -- one good inning and one bad one," Wedge said. "Unfortunately, we had to go to our bullpen early, and that made it tougher to get to the end."
Detroit's Zach Miner gave up five runs in five innings.
Detroit took the lead on Ryan Raburn's bases-loaded walk in the fourth, and Everett followed with a liner just inside the left-field foul pole for his first homer with the Tigers.
Miner almost got out of his own bases-loaded jam in the fifth, but Raburn slipped and fell while chasing Asdrubal Cabrera's fly to deep left. Raburn got up and nearly made a running catch, but the ball hit off his glove for a three-run double.
"It didn't seem wet out there, but maybe I hit the one wet spot -- I just know my feet went out from me," Raburn said. "Even then, I thought I could get to the ball, but it kept going. I thought that was a popup, and it went all the way to the wall."
Everett started a rally in the sixth. He reached on a nice bunt single, took second on a sacrifice and moved to third on a groundout before Magglio Ordonez singled him in.
The Indians tied it again in the seventh on Martinez's broken-bat RBI double off Bobby Seay. Martinez moved up on a wild pitch. After Choo walked, Zumaya came in and gave up a tiebreaking sacrifice fly to DeRosa, but that was the end of Cleveland's offense.
"I didn't want to give up that run, but after that, everything went exactly the way I wanted it," Zumaya said. "That's a good feeling."
The Tigers announced C Matt Treanor had surgery Thursday for a bone spur in his hip, a move that is expected to end his season. Treanor has been sidelined since April 24. ... Everett's previous grand slam came in 2003. ... The Indians made three roster moves after Friday's win, putting RHP Joe Smith (rotator cuff) on the 15-day disabled list, optioning OF Trevor Crowe to Triple-A Columbus and designating INF Tony Graffanino for assignment. They recalled INFs Josh Barfield and Luis Valbuena and OF Matt LaPorta from Columbus.