The Rangers trailed 3-0 entering the game's late stages, but RBI from Ian Kinsler in the sixth and eighth innings narrowed the deficit to one run.
Then in the ninth, Pierzynski led off the frame with a score-tying solo shot against Angels closer Ernesto Frieri (0-3).
Two outs later, Soto curled a pitch just inside the left-field foul pole for a walk-off, game-winning homer.
Jason Frasor (1-2) was credited with the victory as Texas broke a four-game skid in the opener of a three-game set.
J.B. Shuck belted a bases-empty homer and Josh Hamilton went 2-for-4 with a two-run single for Los Angeles, which fell for a fourth consecutive time.
2014: NFL Cheerleaders [PHOTOS]
Michael Sam cut from the Dallas Cowboys