Gennett tallied three RBI and Khris Davis homered in the win, which improved Milwaukee to 5-3 on their road trip and snuffed out the Rangers' eight-game winning streak.
Milwaukee starter Marco Estrada (5-4) got his first win since May 24. Estrada held Texas to one run on four hits and did not issue a walk in six innings.
Mitch Moreland led Texas with a solo home run and Leonys Martin had a pair of hits. Alexi Oganda (5-4) was charged with the loss after allowing four runs on six hits in 6 1/3 innings.
Gennett homered in the first inning to give Milwaukee a 1-0 lead and again in the fifth to make it 3-1. Moreland homered in the third for the only Rangers run.