The Indians seemed in control of the contest after they had forged an 8-5 lead in the top of the 10th thanks to back-to-back homers from Travis Hafner and Ezequiel Carrera.
But the Tigers came right back in the next half-inning against Indians closer Chris Perez (0-3) -- all with two out.
Alex Avila and pinch-hitter Andy Dirks began the rally by drawing walks and Austin Jackson added a double for an RBI. Omar Infante (4-for-6, three RBI) followed with a soft liner into center field to chase home Dirks and Jackson for the game-tying runs.
That brought up Cabrera, who smacked a pitch into the center field bullpen to give the Tigers a come-from-behind victory and a three-game series sweep.
Jackson finished with four hits Darin Downs (1-0) earned the victory for the second-place Tigers, who stayed 1 1/2 games behind the American League Central-leading Chicago White Sox.
Carrera, Hafner and Michael Brantley each had three hits for Cleveland, which lost its ninth straight.
2014: The Year in Music [PHOTOS]