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.
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.
"I'm super excited," Clevinger said Monday of playing for the Padres. "It's the most exciting team in baseball by far right now. It's kind of the place to be right now."
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 523⅓ 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. Clevinger said Monday that he doesn't want this one mistake to define who he is as a player and teammate.
"I wasn't gonna let that moment define me as a person going forward," he said. "I know the changes that had to be put in place and maybe some self-reflecting that needed to be done was done, and I never wanna put any organization let alone the Indians in a predicament like that again. I never was a distraction before and I don't plan on ever being a distraction from anybody."
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 his two starts.
ESPN's Jesse Rogers and The Associated Press contributed to this report.