Rogers (2-1) stymied the Pirates on three hits, working around three walks and striking out five while making his eighth major league start.
Gomez launched his shot in the fourth inning, giving the Brewers a 3-0 lead, and Aramis Ramirez added a two-run clout in the fifth for Milwaukee.
Erik Bedard (7-14) was tagged for six runs on six hits through 4 2/3 frames as the Pirates fell for the sixth time in eight games.
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