With the score tied 1-1 in the eighth, Craig Gentry drew a leadoff walk from Carter Capps (0-1) and went to second on Leury Garcia's groundout.
One out later, Andrus delivered a hit to right field for decisive blow.
He later stole second and came around to score on Adrian Beltre's single.
Nelson Cruz also provided an RBI as the Rangers won for the fifth time in seven contests.
Reliever Joe Ortiz (2-0) earned the decision with two scoreless innings.
The Mariners, meanwhile, fell for the fourth time in five games. Jason Bay knocked in their only run.
Notable deaths of 2014 [PHOTOS]