With the contest tied 3-3, Don Kelly led off the sixth with a double, went to second on Ramon Santiago's sacrifice and scored when Jackson grounded the first pitch he saw from Rays starter Alex Cobb (3-5) up the middle for a hit.
Prince Fielder added an insurance run with another RBI single off reliever Jake McGee later in the frame for Detroit.
Drew Smyly (3-3) earned the victory with five solid innings, holding the Rays to three runs on three hits, while four Detroit relievers tossed one scoreless frame each to nail down the triumph.
Jose Valverde picked up his 15th save as the Tigers claimed three wins in their four-game set at Tropicana Field.
Cobb ended up allowing five runs (four earned) on eight hits through 5 1/3 innings for the Rays.
Hideki Matsui went 2-for-3 with an RBI in the loss.
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