Steven Kwan also homered in manager Terry Francona’s 1,927th win, breaking a tie with Casey Stengel for 13th on the career list.
Civale (5-2), a potential trade target ahead of Tuesday’s deadline, pitched six innings in his second straight win. He allowed three hits, struck out four and walked two.
“Early on his curveball was really working and then he got comfortable with his cutter and threw his fastball,” Francona said. “He just really knows how to pitch.”
The 28-year-old right-hander is 3-0 with a 2.08 ERA in his last seven starts.
Enyel De Los Santos, Trevor Stephan and Emmanuel Clase each pitched an inning after Civale departed, completing a four-hitter.
Cleveland (53-53) earned a split of the four-game series.
White Sox right-hander Michael Kopech allowed three runs and four hits in five innings. He struck out two and walked five.
Kwan led off the fifth with his fifth homer of the season. After Andrés Giménez struck out swinging, Ramírez hit a drive deep to right field for his 17th homer on the year.
Following a two-out walk by Giménez in the seventh, Ramírez hit another shot to right to give Cleveland a 5-0 lead.
“A game like today is the game everyone wants to get,” Ramírez said through a translator. He added he talked Francona out of giving him a day off.
“The days I have off are the days we’re not playing.”
The Guardians got their first run when Giménez walked with the bases loaded with two outs in the second. Kopech issued four walks and threw a wild pitch in the inning.
“Walks got him,” White Sox manager Pedro Grifol said of Kopech, who has just one win in his last 10 starts.
Chicago (43-64) had runners at second and third with one out in the sixth but Civale escaped after inducing a pair of flyouts.
Chicago pitching coach Ethan Katz said the club’s fifth starter likely will come from the current roster. Relievers Tanner Banks and Jesse Scholtens could get chances after the White Sox traded Lucas Giolito and Lance Lynn to contenders.
Guardians: RHP Cal Quantrill (shoulder inflammation) threw a bullpen session. He has been on the injured list since July 6.
White Sox: C Seby Zavala needed a moment with a trainer after catching a bouncing pitch under his hockey-style mask in the fifth. He remained in the game.
Guardians: Noah Syndergaard (1-4, 7.16 ERA), acquired last week in a trade with the Los Angeles Dodgers, makes his Guardians debut Monday night at Houston against fellow RHP J.P. France (6-3, 2.87 ERA).
White Sox: The White Sox haven’t named a starter for Tuesday night’s series opener at Texas. RHP Jon Gray (6-5, 3.66 ERA) is scheduled to go for the AL West-leading Rangers.
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