The White Sox had grabbed a 3-2 lead in the top of the 10th on Conor Gillaspie's homer off Tommy Hunter (4-3).
Henry Urrutia began the comeback with a one-out, pinch-hit single against White Sox closer Addison Reed (5-3). Chris Dickerson came in to run, and went to third when Nick Markakis dumped a single into left field.
Markakis stole second as Nate McLouth struck out for the second out. That brought up Wieters, who delivered a solid hit down the right field line to score both runners for the win.
Dayan Viciedo homered for Chicago, which dropped a ninth straight.
Notable deaths of 2014 [PHOTOS]