Hamilton added an RBI single during a three-run third inning during which Ian Kinsler launched a shot off the left-center wall for a two-run double.
Yu Davish (12-8) broke a personal three-start winless streak, holding the Tigers to three runs on six hits over 6 2/3 frames.
The Japanese right-hander walked three and struck out five as American League West-leading Texas took two out of three from the Tigers.
Detroit's Rick Porcello (9-7) was torched for six runs (four earned) on nine hits in six innings, absorbing the loss.
Miguel Cabrera drove in two runs for the second-place Tigers, who fell two games behind the American League Central-leading Chicago White Sox.