Clevinger (4-4) allowed just two singles, walked none and struck out three against his original team after missing six weeks with right biceps inflammation. He had last pitched on June 14 against the Dodgers in Los Angeles, and he left after 72 pitches in this one.
Clevinger and manager Pedro Grifol said the righty's stuff was lively in a bullpen session on Monday.
"He just carried it out here today," Grifol said. "He had adrenaline. His fastball was good. It had late life on it. His changeup was good. He mixed in some breaking balls."
The underperforming White Sox began their pre-trade-deadline sell-off this week, sending two starters and two relievers to the Los Angeles teams. Lucas Giolito and Reynaldo Lopez went to the Angels on Wednesday, with Lance Lynn and Joe Kelly shipped to the Dodgers on Friday.
Clevinger, 32, has a 3.88 ERA, 56 strikeouts and 26 walks in 13 starts over 67⅔ innings during his first season with Chicago.
He has undergone two Tommy John surgeries in his career (2012, 2020), which covers five years with Cleveland (2016-20), two with San Diego (2020, 2022) and the current campaign with Chicago.
Navarro, 25, is 3-1 with a 3.32 ERA, two saves and 39 strikeouts over 35 combined relief appearances between Charlotte and Double-A Birmingham. This is his first time on a major league roster since the White Sox signed him as an international free agent on March 10, 2018.
Shaw, a 35-year-old journeyman, made six relief appearances with the White Sox this season and had a 9.39 ERA in 7⅔ innings pitched.
The Associated Press and Reuters contributed to this report.