Cleveland opened a seven-game lead over Chicago in the AL Central with its 15th win in 18 games. The Guardians (83-67) can clinch the franchise's first division title since 2018 as early as Sunday.
Bieber (12-8) earned his fourth straight win. The 2020 AL Cy Young Award winner allowed seven hits, struck out five and walked none in 7 2/3 innings.
''Like a lot of good pitchers, he kind of worked into finding his stuff,'' Guardians manager Terry Francona said. ''He kind of pitched himself into looking like Bieber, which is pretty impressive.''
Trevor Stephan finished the eighth and Emmanuel Clase handled the ninth for his AL-best 37th save in 41 chances.
Cleveland's pitching staff issued just one walk in the three-game set.
''We're a confident bunch and we're having a lot of fun doing it,'' Bieber said.
Gavin Sheets homered for Chicago in the eighth. Johnny Cueto (7-9) allowed four runs, three earned, and eight hits in six innings in his third straight loss.
Chicago isn't out of it yet, but it needs lots of wins - and help - down the stretch.
''(The season) doesn't come down to three games but we knew what we had to do this series and we weren't able to do it. It's frustrating,'' Sheets said. ''It's not just this series; we had to play better all season.''
The White Sox (76-74) have dropped three straight and four of six overall.
Cleveland went ahead to stay on Jose Ramirez's sacrifice fly in the third. Steven Kwan hit an RBI triple in the fifth and scored on another Ramirez sacrifice fly, giving the Guardians a 4-1 lead.
The White Sox, who started September with an 8-2 streak but failed to make up much ground on the streaking Guardians, got a run in the first on Eloy Jimenez's RBI single.
The Guardians' second run was set up by a Cueto pickoff attempt that bounced in front of first baseman Jose Abreu's glove in the third. The ball rolled to the dugout fence and allowed Amed Rosario to dash from first to third. He came home a pitch later on Ramirez's sacrifice fly.
Guardians: RHP Zach Plesac (broken finger) could get a start or two before the end of the season, but Francona said it only will happen if Plesac is ready. He has been out since Aug. 30.
Guardians: Cody Morris (0-2, 2.30 ERA) faces Texas right-hander Jon Gray (7-7, 3.80 ERA) on Friday night.
White Sox: RHP Lucas Giolito (10-9, 5.07 ERA) goes against Tigers LHP Eduardo Rodriguez (3-5, 4.35 ERA) on Friday night.
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