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.
Beautician charged with giving client fatal silicone butt injection
Ohio bar shooting arrested, charged with murder