Vargas (7-5) blanked the Mariners over the first six innings, then yielded a run in the seventh before exiting. The left-hander allowed eight hits and a walk while striking out five.
Trout got the Angels off to a quick start with a two-run blast in the first inning.
J.B. Shuck went 3-for-5 with a run scored for Los Angeles, which got RBI from Kole Calhoun and Chris Nelson in clinching its three-game series.
Seattle's Erasmo Ramirez (4-1) surrendered three runs (two earned) on seven hits in a 6 2/3-inning effort.
Nick Franklin got the Mariners' only RBI.