The Tigers let a pair of one-run leads get away before coming from one run behind in the ninth to hand the White Sox their third straight loss.
Matt Thornton (1-1) came on in the eighth to get the final out of the inning and preserve a one-run Chicago advantage, and he took the mound again in the ninth.
Ramon Santiago led off the ninth with a bunt that came back to Thornton and was turned into an out. Alex Avila, however, reached on a single and Peralta, after fouling off the first three pitches he saw, hit his first homer of the season to right.
Peralta's homer made a winner out of Jose Valverde (2-0), who worked a scoreless top of the ninth.
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