Verlander (4-3) was erratic early, allowing three runs in the first two innings. He settled a bit after that, but the Indians took a 6-1 lead before Detroit closed the gap with a four-run seventh.
It was 7-5 when Perez allowed an unearned run in the ninth, but with men on first and second, Cabrera hit a weak grounder to third to end it.
It was Perez's sixth save.
Jimenez (3-2) allowed a run and three hits in six innings, striking out eight with one walk. He's won three straight starts for the first time since the Indians acquired him in a trade in 2011.
Jhonny Peralta hit a solo homer for Detroit.
Perez got the first out of the ninth, but first baseman Nick Swisher's error on a throw to first started a Detroit rally. Omar Infante followed with a single, but second baseman Jason Kipnis fielded Austin Jackson's grounder up the middle and flipped to shortstop Asdrubal Cabrera for a force.
Torii Hunter's RBI single brought Cabrera to the plate, and the count went full on the Triple Crown winner before his slow grounder was fielded cleanly for the final out.
Swisher opened the scoring with an RBI double in the first, and Jason Giambi drew a bases-loaded walk to make it 2-0.
The Detroit ace got the first two outs of the second but allowed another run on a single by Kipnis and a double by Asdrubal Cabrera.
Verlander allowed six hits and five walks in five innings, striking out seven and leaving after 110 pitches. After Peralta's homer got the Tigers on the board in the third, Cleveland added an unearned run off Verlander in the fifth to make it 4-1.
Swisher walked to lead off the inning, and Carlos Santana hit what looked like a double play grounder to first baseman Prince Fielder, who stepped on first and threw to second to trap Swisher between bases. Miguel Cabrera, the third baseman, was involved in the rundown because a defensive shift had him playing close to second, and he threw the ball away for an error. Swisher eventually scored on a two-out single by Mark Reynolds.
Asdrubal Cabrera's RBI single in the sixth made it 5-1, and pinch-hitter Mike Aviles added a run-scoring single of his own in the seventh, but the Cleveland bullpen almost gave up the lead. Jimenez left after 93 pitches, and Nick Hagadone allowed a double and two walks to start the Detroit seventh.
Cody Allen came on and allowed a sacrifice fly to pinch-hitter Brayan Pena. Infante added a two-run triple and scored on Jackson's groundout to make it 6-5.
A sacrifice fly by Giambi in the eighth gave Cleveland an insurance run the Indians would end up needing. Detroit's Matt Tuiasosopo hit into an inning-ending double play with two on in the bottom of the eighth.
When Jimenez was acquired two years ago, the Indians hoped they were getting an ace who could help them compete with teams like the Tigers. The right-hander hasn't come close to expectations, but he's pitching better now.
This was Cleveland's seventh game this season against a former Cy Young Award winner, and the Indians are 6-1 in those games.
NOTES: It rained a bit early in the game, but not hard enough to halt play. ... Peralta and Miguel Cabrera extended their hitting streaks to 10 games. ... Detroit's Prince Fielder was hit in the left thigh in the eighth and went hobbling toward the mound, but he wasn't charging reliever Joe Smith. He was just trying to run through the pain a bit and his momentum took him in the direction of the pitcher. ... Jimenez's previous three-start winning streak was in July 2011, less than a month before the trade to Cleveland. ... The Indians have not lost any of their last six series. The rubber game of this three-game set is Sunday, with Cleveland's Zach McAllister (3-3) facing Detroit's Rick Porcello (1-2).