Clinging to a slim 2-1 lead entering the frame, the Rays exploded for five runs to put the game away.
After Tampa Bay scored a run when Twins pitcher Nick Blackburn (7-6) threw wildly to first on Jason Barlett's sacrifice bunt, Carl Crawford was intentionally walked with one out to load the bases. But Longoria crossed up the strategy by grounding a two-run single up the middle to make it 5-1.
Brian Duensing came in to relieve and surrendered a pinch-hit, two-run double to Kapler, and the Rays went on to take three games of a four-contest set at Target Field.
James Shields (7-8) pitched well enough to win, allowing four runs on eight hits over six innings, and Rafael Soriano picked up his 21st save.
Jim Thome, who socked two homers Saturday, went 2-for-4 with two RBI as the Twins dropped their first homestand in their new ballpark, going 3-4 in the seven-game stretch.