Tigers starter Phil Coke had allowed one run on two hits over 3 1/3 innings when he turned an ankle and had to be taken out of the game. He was replaced by Furbush, who tossed 3 2/3 innings of scoreless relief to pick up the victory.
Furbush, a highly-touted prospect who was leading the International League in strikeouts, was called up from Class AAA Toledo last week when reliever Brad Thomas was placed on the disabled list.
Victor Martinez and Jhonny Peralta each drove in a pair of runs, Andy Dirks hit his first major league homer and Miguel Cabrera delivered a tie-breaking single for the Tigers, who began a three-game series against the Rays with a victory.
Jeremy Hellickson (5-3) absorbed the loss, yielding two runs on four hits over 6 1/3 frames.
Sam Fuld drove in a pair of runs for the Rays.