The Red Sox went ahead 5-4 in the eighth on Dustin Pedroia's solo homer, but the Twins forced extra innings when Brian Dozier answered a bases-empty clout in the ninth.
The game stayed knotted until there were two outs in the 11th, when Saltalamacchia reached on an infield hit against Jared Burton (0-1) and went to second on Will Middlebrooks' sharp single to left-center field.
That brought up Drew, who pounded a pitch off the base of the left field "Green Monster" wall, sending Saltalamacchia home with the decisive run.
Drew finished 4-for-5 with three RBI while Pedroia and Shane Victorino each went 3-for-5 with a solo shot for Boston, which took the opener of a four-game series.
Clayton Mortensen (1-2) fired 2 1/3 scoreless innings to get credit for the win.
Justin Morneau drove in a pair of runs for the Twins.