Moore surrendered a three-run homer to David Ortiz in the first inning, but followed that with five scoreless frames in notching his seventh win in eight starts to begin the season.
The left-hander yielded three hits, walked two and struck out eight.
Jake McGee, Josh Lueke and Joel Peralta combined for two innings of shutout relief and Fernando Rodney struck out the side in the ninth for his seventh save as the Rays claimed the first of a three-game series.
Red Sox starter John Lackey (1-4) was roughed up for five runs on nine hits in a 4 1/3-inning start.
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