In his second start with Chicago since being acquired from San Diego on July 31, Peavy (2-0) allowed six hits with two walks and eight strikeouts. The 2007 NL Cy Young winner spent three months on the disabled list with an ankle injury and his return was slowed when he was hit in the elbow by a line drive during a minor league rehab game.
The White Sox won despite getting only two hits.
Bonine (0-1), in his fourth stint with Detroit this season, didn't allow a hit until Beckham's two-run homer with two outs in the sixth followed a costly error on Tigers third baseman Brandon Inge.
With one out, Alex Rios reached when Inge bobbled his grounder and threw late to first
- only the second error by the Tigers in the last 22 games.
After Scott Podsednik was thrown out when Bonine deflected his grounder to second baseman Placido Polanco, Beckham drove his 14th homer to deep left to give Chicago a 2-0 lead.
Matt Thornton worked the ninth for his second save.
Bonine went 6 2-3 innings, leaving after walking Jermaine Dye and giving up a single to Carlos Quentin.
The White Sox, the defending AL Central champions, were eliminated Thursday night when the Tigers beat Cleveland 6-5 in a game that saw Inge make a game-ending diving catch of a line drive.
Bonine faced the minimum 16 batters through the first four innings despite two walks because he got A.J. Pierzynski to hit in a first-inning double play and Rios was thrown out trying to steal in the third.
Bonine was making his third start of the season. He got a no-decision against the White Sox in July, pitching six innings and giving up seven hits and three runs.
