Justin Verlander survived one rough inning to pick up the win in the opener of a series between the top two teams in the American League Central.
The home run by Martinez ignited a three-run second inning that put the Tigers in front for good. Detroit added runs in that inning on a single from Torii Hunter and a bases-loaded walk to Miguel Cabrera.
Martinez also scored in the fifth on a double play grounder off the bat of Brayan Pena and Pena singled Martinez home in the seventh with the final Detroit run.
Verlander (8-4) allowed three runs on seven hits with six strikeouts in seven innings. Cleveland scored three times in the fifth, two of the runs coming in on a double by Matt Kipnis.
The loss went to Ubaldo Jimenez (4-4), who survived just three innings.