Grimm (3-3) broke a personal three-start losing streak, yielding a pair of runs on seven hits over 6 2/3 innings. The right-hander walked one and struck out two.
Andrus delivered an RBI triple in the second inning, igniting a three-run outburst, and added a run-scoring hit in the third while Mitch Moreland homered and Lance Berkman contributed two RBI for Texas.
The win was the fourth in five games for the American League West- leading Rangers.
Tigers starter Anibal Sanchez (4-4) was reached for six runs (five earned) on nine hits, lasting just 2 2/3 innings.
Pregnant Mila Kunis wins 'Best Villain' at MTV Movie Awards
'Happy' fan videos make Pharrell cry [VIDEO]