Cleveland leads the AL Central by one game over Minnesota, while the White Sox moved within 2 1/2 games of the Guardians. Chicago had lost three in a row.
''Every time I pitch now is kind of like a playoff game because we're in the race,'' Cueto said. ''I know what I have and I like to compete.''
Cueto (6-5) posted his 10th consecutive quality start, the first double-digit string by the White Sox since Carlos Rodon in 2016. The right-hander struck out two and walked one in beating Cleveland for the fifth time in five career decisions.
Manager Tony La Russa pulled Cueto for Liam Hendriks with one on and two outs in the ninth after 113 pitches. Hendriks struck out rookie Oscar Gonzalez for his 28th save.
The 36-year-old Cueto nearly pitched his first shutout since 2016 with San Francisco.
''Cueto is an artist and he was clutch for us,'' La Russa said. ''He looked the same at the end as he did at the beginning, but the only guy he wasn't going to face was Gonzalez. Liam is happy that we shared the wealth.''
Shane Bieber (8-7) had his four-start winning streak snapped, allowing two runs on five hits in 6 1/3 innings. The 2020 AL Cy Young Award winner struck out four in his first loss since July 24 at the White Sox.
Abreu opened the scoring in the sixth by doubling to straight-away center, driving in Luis Robert. Chicago only had one hit off Bieber entering the inning.
''Johnny has been a great arm for a long time,'' Bieber said. ''But obviously it sucks to come out on the other end of a game like this.''
Elvis Andrus extended the White Sox's lead to 2-0 on an unusual play in the seventh, scampering home after Will Benson's throw from right field bounced and struck catcher Luke Maile on his mask.
Benson caught Josh Harrison's flyout, but was charged with an error when the ball dislodged Maile's mask and bounced away from him. Reliever James Karinchak was unable to retrieve it before Andrus touched the plate.
''(Andrus) is a hell of a threat and he takes pride in his baserunning,'' La Russa said. ''He still has his legs.''
White Sox catcher Yasmani Grandal injured his left knee while attempting to score from second base in the seventh. Grandal, who was thrown out by Myles Straw, was unable to put any weight on his leg as he was helped off the field.
Grandal was on crutches in the clubhouse and said he wouldn't know the specifics of his injury until undergoing tests. La Russa said he will be placed on the injured list Sunday.
Andrus went 1 for 4 in his second game with the White Sox. The 33-year-old shortstop signed as a free agent after being released by Oakland.
Oscar Gonzalez had two of Cleveland's hits, along with its lone walk, and rookie Steven Kwan, Josh Naylor and Maile each had one.
The Guardians have won 10 of their last 14 and are 23-14 since July 10.
The first pitch was delayed by 2 hours, 55 minutes due to rain and started at 9:05 p.m. Cleveland has played an MLB-high 10 doubleheaders and had 11 games postponed, including three against Chicago.
''A three-hour rain delay isn't good to begin with, but I think it's even harder on starting pitchers,'' Bieber said. ''That's why Johnny and I gave each other a look when we finally got out there.''
Robert started for the first time in eight games after spraining his left wrist sliding into second base on Aug. 12.
Chicago's Andrew Vaughn, who was struck on the face by a pitch from Trevor Stephan that first glanced off his shoulder Friday, was in uniform but did not play.
White Sox: OF Eloy Jimenez (right knee hyperextension) batted third and served as the designated hitter, one night after exiting in the middle of an at-bat in the eighth inning. Jimenez said he responded well to treatment after being injured on a hard swing.
Guardians: C Austin Hedges (right ankle sprain) was held out of the lineup by manager Terry Francona, but was available. Hedges slid feet-first down the steps of the dugout in the fifth inning Friday while pursuing a wild pitch, forcing him to leave the game.
White Sox: RHP Dylan Cease (12-5, 2.09 ERA) starts the three-game series finale. Cease leads the majors with 11.99 strikeouts per nine innings, ranks third in ERA and is fourth with 178 strikeouts.
Guardians: RHP Aaron Civale (2-5, 5.63 ERA) is on a seven-start winless streak, losing twice with a 3.55 ERA over 33 innings since June 21. Civale also has taken four straight no-decisions since July 8.
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