Morneau's 14th homer of the season was a solo drive in the third off Robinson Tejeda and it gave Minnesota a 3-to-1 lead.
Bonser allowed two runs and six hits in five innings, striking out six and walking two.
Twins closer Joe Nathan got three outs for his ninth save in ten chances as Minnesota finished its road trip 4-and-5.
Tejeda gave up five runs and three hits in three-plus innings, throwing 86 pitches in his shortest outing of the season.
Sammy Sosa pinch hit for Texas with two on and two outs in the ninth, but he struck out, remaining two home runs shy of 600.