Grimm (2-0) has joined the Texas rotation after injuries affected two of the team's starters and he gave up five hits with four strikeouts and no walks.
His teammates built a 3-0 lead through eight innings and Ian Kinsler doubled home a fourth Texas run in the top of the ninth that turned out to be crucial.
Joe Ortiz came on to pitch the ninth for Texas and put two runners on with one out before giving way to closer Joe Nathan.
Nathan struck out Trevor Plouffe, but gave up a three-run homer to Oswaldo Arcia. Ryan Doumit then flew out to center to end the game and give Nathan his eighth save.
Scott Diamond (1-2) allowed 12 hits and three runs in 6 1/3 innings to take the loss.
Texas has won seven out of eight. Minnesota has dropped three in a row following a five-game winning streak.