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 José Ramirez’s sacrifice fly in the third. Steven Kwan hit an RBI triple in the fifth and scored on another Ramírez 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 Jiménez’s RBI single.
The Guardians’ second run was set up by a Cueto pickoff attempt that bounced in front of first baseman José 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 Ramírez’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.