Leury Garcia had two singles and a game-tying sacrifice fly in the ninth for the White Sox, who reached .500 for the first time since April 21.
Rafael Devers had an RBI double for Boston, which has lost 13 of 17. The Red Sox have lost six straight series.
Boston manager Alex Cora was ejected by home plate umpire Carlos Torres for arguing after Trevor Story was called out on strikes.
Abreu drove in automatic runner AJ Pollock and Luis Robert followed with an RBI single off Matt Barnes.