Doubront (9-6) allowed one run on five hits, walking one and striking out three. The left-hander bounced back after a seven-run shellacking at the hands of the New York Yankees in his previous start last week.
Stephen Drew cranked a three-run homer and Will Middlebrooks added a two-run shot while Jonny Gomes and Jarrod Saltalamacchia each drove in a pair of runs for Boston, which took 2-of-3 games in the series.
Giants starter Barry Zito (4-9) was lit up for six runs on seven hits, lasting only 3 2/3 innings.
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