Bauer (6-5, 2.50) continued to pile up strikeouts, moving into a tie for second place on the American League leaderboard with eight whiffs. His effort led the Indians to their sixth win in eight games against the Sox.
He matched Boston's Chris Sale with 129 strikeouts, trailing AL leader Gerritt Cole of Houston. Cole entered his start Monday against Tampa Bay with a league-high 130.
Manager Terry Francona said Bauer has been able to maintain his velocity and movement, and actually get better as games wear on.
"A lot of pitches get through that third time through the order and you start to worry a little bit," Francona said. "But because he hasn't been going to so many deep counts and his stuff gets better, he's every bit as effective."
Bauer fell short of extending his streak of double-digit strikeout starts, but picked up the fifth win in his last seven outings. He threw 100 pitches and lowered his earned-run average to 2.50 on the season. Bauer entered the game with at least 10 strikeouts in five of his last six appearances.
"He works so hard to get ahead and throw strike one and then put guys away," catcher Roberto Perez said of Bauer. "He's pitching his tail off and hopefully he keeps doing that the rest of the way."
Heavy rain forced the delay that ended Bauer's outing in the seventh. It was Bauer's second consecutive start against Chicago after he took a tough-luck loss Wednesday in a game that saw him strike out 12 and allow just three runs in 7 2/3 innings.
He was even better on Monday, taking advantage of two double plays in the first three innings to keep Chicago's offense from ever mounting a serious threat.
Bauer pushed Francona for the chance to finish the game following the delay. He joked afterward that pitching coach Carl Willis was lucky he didn't have to deliver the news that the right-hander was not going back out for the eighth.
"I told him anyone but Tito and I was throwing fists," Bauer said with a grin. "I'm sitting there in the dugout thinking, 'You know, the hard stuff is not going to come down for quite a while.'"
Right-hander Evan Marshall relieved Bauer after the rain delay in the eighth, allowing a run on two hits before leaving with right elbow soreness. Dan Otero pitched the ninth inning for Cleveland, allowing a solo home run to Chicago's Matt Davidson.
Jason Kipnis drove in a pair of runs on three hits, including a solo home run in the fifth inning against White Sox starter Dylan Covey (3-2, 2.90) that gave the Indians a 5-0 advantage.
Kipnis raised his season batting average to .208, including five home runs and 27 RBI. He entered the game hitting .195 in the month of June, but his effort at the plate caught the attention of his starting pitcher.
"Kip's year hasn't gone the way he's wanted so far, but his mind-set and his attitude and stuff like that has been awesome" Bauer said. "So, I'm really happy that he had a night like tonight."
Cleveland got on the scoreboard with three runs in the second inning thanks to a White Sox error and a vanishing act as Perez made the baseball disappear into the right field wall. Perez drove a Covey pitch to right for a bases-loaded ground rule double that brought home Lonnie Chisenhall and Rajai Davis with the first two runs of the game. The ball disappeared on the fly between the wall pad and a chain link fence that separates fans from field level.
Cleveland added another run in the third when Kipnis drove in Davis with an RBI single to right. Davis had reached on a fielder's choice and taken second on an error when Sox first baseman Jose Abreu couldn't handle Covey's pickoff throw.
The Indians batted 1-for-3 with a walk, a strikeout and three RBI in four opportunities with the bases loaded. Chisenhall forced home a run in the sixth when he drew a bases-loaded walk against Chicago reliever Hector Santiago.
Rain or shine
Monday's game entered a rain delay at 9:35 p.m. with the Indians batting in the bottom of the seventh inning. Crews pulled the tarp and play resumed after a 35-minute delay. The last time a game at Progressive Field was shortened by rain was Sept. 20, 2013 against Houston in the top of the seventh inning with the Indians in front, 2-1.
What it means
Cleveland has won 18 of its last 24 games against the White Sox since April of 2017, and improved its record to 6-2 against the South Siders this season. The first-place Tribe picked up a half-game in the American League Central Division standings over the second-place Tigers, who were idle. Cleveland's lead is now three games in the division.
Bauer threw 100 pitches, 59 (59 percent) for strikes. Covey threw 94 pitches, 56 (59 percent) for strikes.
Thanks for coming
The White Sox and Indians drew 17,271 fans to Progressive Field on a steamy Monday evening. First pitch was at 7:10 p.m with a temperature of 89 degrees.