Shin-Soo Choo added a three-run double for the Rangers, who won the final two games of this three-game set. Martin finished with two hits and scored three runs for Texas, which has won eight of its last 11 games.
Josh Willingham homered and drove in three runs for the Twins, who have dropped six of their last seven.
The Rangers broke a 4-4 tie in the top of the eighth when Martin led off the inning against Minnesota reliever Casey Fien with a double, moved to third on a Robinson Chirinos sac bunt and scored on Odor's sac fly.
Fien (3-2) suffered the loss in one inning of work.
Texas got going in the second, as Choo's three-run double staked the Rangers to the early lead.
Minnesota cut the deficit to a run in the third when Trevor Plouffe and Willingham each came through with RBI singles and took the lead in the fifth on Willingham's two-run homer.
The Rangers tied the game in the sixth on Chirinos' run-scoring single.
Texas starter Nick Martinez was charged with four runs on nine hits in 5 2/3 innings, while his counterpart, Minnesota's Samuel Deduno, was also hit for four runs on nine hits in 5 1/3 innings.