Tony Abreu then dropped a single into left field to score Francoeur with the tying run, with Hosmer thrown out at third.
Jarrod Dyson, however, sent Abreu to third with a single and Alex Gordon chased Peavy with a solid RBI hit to right-center field.
Billy Butler sealed the win with a two-run homer in the eighth and Kansas City held on to even its three-game series.
Bruce Chen (8-9) corralled the victory, holding the White Sox to two runs on seven hits over 6 2/3 innings.
Kelvin Herrera tossed 1 1/3 frames of scoreless relief and Greg Holland turned in a 1-2-3 ninth for his second save.
Gordon Beckham homered for the American League Central-leading White Sox, whose lead over second-place Detroit was shaved to one-half game with only their second loss in seven games.