Evan Longoria's two-run homer in the top of the ninth put Tampa Bay ahead 6-4, but the White Sox weren't done. Abreu's drive off closer Grant Balfour (0-1) won it after Chicago pitchers combined for 11 walks.
Abreu, the rookie slugger from Cuba who signed a $68 million, six-year contract in the offseason, finished with three hits and six RBIs.
Adam Eaton hit a grounder to second base, but Tampa Bay was unable to turn the double play after getting a force at second.
Rays manager Joe Maddon challenged the call, but Eaton clearly beat the throw to first and the ruling was upheld on replay. Balfour walked Marcus Semien to load the bases against and Abreu drove an 0-1 pitch to right-center for his ninth home run.
Matt Joyce drew his team-record fifth walk of the game leading off the ninth against White Sox reliever Matt Lindstrom (2-1). Longoria followed with his third home run of the season to center to break a 4-all tie.
Tyler Flowers had two hits and two RBIs for the White Sox in the opener of a four-game series. Chicago starter Erik Johnson couldn't get out of the second inning.
The Rays tied a club mark by drawing 11 walks.
Tampa Bay starter Chris Archer pitched six-plus innings. He allowed four runs and nine hits.
Johnson was chased after walking Joyce with the bases loaded on his 39th pitch of the inning, giving the Rays a 4-1 lead. The Rays batted around against Johnson in the second.
Abreu hit a solo shot in the third and the White Sox tied it in the fourth when Flowers lined a two-run single with the bases loaded. But a baserunning gaffe by Chicago prevented a potentially bigger inning.
Eaton grounded back to Archer, who caught De Aza between third and home. De Aza was eventually tagged out in a rundown, then Flowers was tagged out while trying to go to third.
The Rays had the bases loaded with one out in the fifth, but failed to score. Hanigan grounded into an inning-ending double play.
After missing five games with a mild left knee strain, Eaton led off the first inning with a single and stole second. He later scored on Abreu's single.
Johnson allowed four runs on four hits and four walks.
NOTES: Before the game, the White Sox claimed reliever Hector Noesi off waivers from Texas. He is expected to throw a bullpen before Saturday's game. ... Rays LHP Cesar Ramos will start Saturday in place of LHP Matt Moore, who had Tommy John surgery on Tuesday.