Pedro Ciriaco followed by hitting a grounder that third baseman Sean Rodriguez threw away, allowing Lavarnway to come home with decisive run.
The designated hitter broke the game open with a two-run double in the next frame and the Red Sox took a second straight win in their four-game set with the Rays.
Felix Doubront (11-9) allowed only one hit but walked five over six full innings en route to the win.
Andrew Bailey secured the victory with his fifth save.
Ben Zobrist went 2-for-4 with four RBI for Tampa Bay.
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