Nick Tepesch (1-0) surrendered three runs on five hits with four strikeouts across 6 1/3 innings to earn the win. Joakim Soria tossed a scoreless ninth for his eighth save of the season.
Robinson Cano stroked a two-run homer and Michael Saunders brought in a run for Seattle, which entered the game with a two-game win streak. Chris Young (3-2) was tagged for four runs on seven hits with three walks in 6 1/3 frames.
After Daniel Robertson got aboard with a walk, Andrus crushed a fastball that barely cleared the left-field wall to give Texas the early lead in the bottom of the first.
Choo and Beltre then singled before Rios plated Choo with a base hit up the middle to make it a 3-0 game. The Rangers later loaded the bases with one out, but Young recorded a strikeout and a groundout to end the opening frame.
Seattle tied the game in the fourth. James Jones hustled on a line drive to the center-field wall for a triple and was brought around on Saunders' single. Cano then drove a two-run shot to center to even the game at 3-3.
Choo put the Rangers ahead for good with a solo homer to lead off the bottom of the fifth.