Cleveland receives right-hander Cal Quantrill, outfielder Josh Naylor and catcher Austin Hedges, along with three minor league prospects — shortstop Gabriel Arias, left-hander Joey Cantillo and shortstop Owen Miller.
The Padres also will receive outfielder Greg Allen and a player to be named later.
The hard-throwing Clevinger represents the front-line starting pitcher long coveted by the Padres, who have won seven of their past 10 games and are five games behind the first-place Dodgers in the National League West.
The deal marks the fifth trade since Saturday for the Padres, who have also acquired first baseman Mitch Moreland from Boston, catcher Austin Nola from Seattle, catcher Jason Castro from the Angels and closer Trevor Rosenthal from Kansas City.
Clevinger, 29, is 42-22 with a 3.20 ERA and 584 strikeouts in just 523 1/3 career innings over four-plus seasons with Cleveland. He was demoted to the minors earlier this season after he and teammate Zach Plesac broke coronavirus-related rules.
A few days after Clevinger and Plesac broke the team’s code of conduct, they drew the wrath of teammates, who expressed their unhappiness during a team meeting.
Led by Fernando Tatis Jr., San Diego is going for its first playoff appearance since 2006. Clevinger now figures to head a talented Padres rotation that also includes right-handers Chris Paddack and Dinelson Lamet.
Clevinger is the third front-line starter traded away by Cleveland in the past year. Trevor Bauer was shipped off to Cincinnati at the 2019 deadline, and two-time Cy Young Award winner Corey Kluber was dealt to Texas in December.
Even without Clevinger, the Indians, who are in the mix for the AL Central title, have plenty of young pitching with ace Shane Bieber, Aaron Civale, Plesac and rookie Triston McKenzie, who has looked like a seasoned veteran in two starts.
ESPN’s Jesse Rogers and The Associated Press contributed to this report.