Ross' two-run blast to left field with one out in the seventh erased a 5-3 deficit, and his solo shot in the ninth off Matt Capps (0-1) handed the Red Sox a late lead.
Alfredo Aceves made the advantage hold up with a scoreless ninth for his third save and the Red Sox notched a win in the first of a three-game set at Target Field.
Daniel Bard (1-2) earned the victory after retiring the final two batters in the eighth.
Jarrod Saltalamacchia also homered for the Red Sox, who banged out 12 hits.
Danny Valencia socked a two-run homer for the Twins.
Minnesota's Jason Marquis surrendered five runs on 11 hits over a 6 1/3-inning start.