Vargas (13-8) won for the sixth time in seven starts overall, holding his opponents to two or fewer runs for the fourth time in five outings.
The M's left-hander issued seven hits and one walk while striking out five as Seattle took the rubber match of a three-game series.
Tom Wilhelmsen collected his 16th save to secure the triumph.
Montero cracked a solo homer in the second inning and added a two-run shot in the sixth to break a 1-1 tie.
The Angels' Jered Weaver (15-2) lost for the first time since May 13 -- a span of 12 starts -- breaking his nine-start winning streak. The right-hander yielded three runs on four hits over seven full frames.
Vernon Wells got the only RBI for Los Angeles, which fell two games off the pace in the American League wild card race.
Celebrity Families of 2014 [PHOTOS]