Vargas (3-3) scattered four hits, walked three and struck out six, corralling his third win in four starts for Los Angeles.
Erick Aybar gave him a quick lead with a bases-clearing double in the first inning while Howie Kendrick went 2-for-4 with a two-run double as the Angels logged just their third win in eight games overall.
White Sox starter Jake Peavy (5-2) surrendered four runs on five walks and four hits, striking out seven.
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