With Seattle trailing 3-2 in the sixth inning, Kyle Seager stroked a one-out double off Nick Tepesch (1-1).
Montero followed by blooping an 0-1 pitch into right field, sending Seager home with the equalizer.
Montero moved to second on a wild throw home by the Rangers' Nelson Cruz, then came home with the go-ahead run when the next batter, Ackley, grounded a ball up the middle for a single.
Rookie Brandon Maurer (1-2) recorded his first major league win, yielding three runs (two earned) on five hits over six innings. The right-hander walked one and struck out five.
Tom Wilhelmsen picked up his fifth save for the Mariners, who split a four-game series at Safeco Field.
Tepesch (1-1) ended up allowing four runs on nine hits over 5 2/3 innings for Texas.
Lance Berkman and Mitch Moreland each supplied an RBI for the Rangers.