Michael Brantley then drew a walk and both runners advanced when Thornton threw wildly to first on a pick-off attempt.
After Jason Kipnes fanned for the second out, Cabrera delivered in the clutch, lining Thornton's first pitch into center field for a two-run base hit.
The rally made a winner of Justin Masterson (4-1), who overcame four walks en route to the victory. The right-hander yielded two runs on four hits while striking out five.
Vinnie Pestano tossed a scoreless eighth and Chris Perez notched his third save as the Tribe claimed the opener of a three-game series.
Conor Gillaspie went 2-for-3 with a solo homer for the White Sox.