Middlebrooks' blast in the fourth inning put the Red Sox ahead 4-2 and Napoli's clout in the next frame sealed the victory as Boston claimed the first of a three-game series.
Napoli also provided an RBI double for the Bosox, who were coming off a Sunday doubleheader loss to Kansas City.
Felix Doubront (2-0) overcame five walks to earn the decision, allowing three runs on three hits over 6 2/3 frames.
Andrew Bailey tossed a scoreless ninth for his fourth save.
Athletics starter A.J. Griffin (2-1) was pounded for nine runs (seven earned) on eight hits over four full frames.
Josh Donaldson and Josh Reddick each supplied two hits and a pair of RBI for Oakland.