The Twins were trailing 4-2 in the eighth when they erupted for a four-run rally against Joel Peralta (2-8) -- all with two outs.
Ryan Doumit belted a solo homer, cutting the deficit to one run, before Peralta issued consecutive walks to Trevor Plouffe and Josh Willingham. That brought up Pinto, who slammed a pitch over the left-center field fence.
Casey Fien (5-2) was credited with the victory and Glen Perkins racked up his 35th save as Minnesota snapped a four-game losing streak.
Wil Myers supplied a two-run homer for the Rays, who came into the contest clinging to a 1 1/2-game lead over Cleveland for the second and final American League wild-card spot.