The Rangers trailed 7-1 after four innings, but scored four runs in the fifth and one run each in the eighth and the ninth to force extra innings.
The Angels went up 10-7 in the 10th on home runs by Chris Iannetta and Albert Pujols, but the LA bullpen could not close the deal. Nelson Cruz homered to lead off the home half of the 10th and Mitch Moreland singled in a run before Andrus came to the plate and delivered the game-winner down the third-base line off Jason Isringhausen (3-1).
Joe Nathan (2-3) was credited with the win. Texas avoided a three-game sweep and ended the day with a two-game lead over the second-place Angels in the AL West.
Andrus tallied three RBI in the game for the Rangers. Kinsler, Moreland and Mike Napoli drove in two runs apiece.
Ian Kinsler had a solo home run in the ninth inning for Texas that tied the score 7-7.
Pujols was 2-for-6 with four RBI for Los Angeles. Torii Hunter and Alberto Callaspo had two RBI apiece.
'SNL': 'Anchorman 2' cast, One Direction sing 'Afternoon Delight' [VIDEO]
Selena Gomez drops F-bomb, walks off stage during Jingle Ball performance