Betancourt connected for a two-run blast in a five-run first inning for the Brewers, who held on to snuff out a five-game losing streak.
Ramirez stroked a solo homer in the seventh inning after the Rangers had closed the gap to 5-3 with a pair of runs in the top of the sixth.
Wily Peralta (3-2) earned the victory for Milwaukee. He allowed three runs – two earned – on five hits over six innings. Jim Henderson was credited with his seventh save.
Rangers starter Justin Grimm (2-2) took the loss after allowing five runs on eight hits and seven strikeouts over five innings. Mitch Moreland hit a solo home run for the Rangers, and Nelson Cruz and Adrian Beltre drove in the other two Texas runs.