The AL Central-leading Indians won the opener 3-2 on Francisco Lindor's tiebreaking single in the ninth inning.
Roberto Perez homered and drove in three runs to back Clevinger (8-5).
Clevinger allowed three hits, struck out six and walked one. In three starts against the Tigers this year — all wins — he has given up one run in 18 innings.
Buck Farmer (3-2) allowed five runs on three hits and three walks in three-plus innings.
Geovanny Urshela hit a two-run double in the second. The Indians scored three more in the fourth, two on a single by Perez.
Perez homered to start a five-run sixth.
The opener was tied at 2 in the ninth when pinch-hitter Jay Bruce tripled with one out off Shane Greene (3-3). Lindor singled through the drawn-in infield.
The Tigers loaded the bases with two out in the ninth inning against closer Cody Allen (2-6), who had allowed the tying run in the eighth.
Indians manager Terry Francona brought in Joe Smith, and he retired Miguel Cabrera on a line drive to Lindor at shortstop for his first save.
Detroit was playing for the first time since trading Justin Verlander to the Houston Astros and Justin Upton to the Los Angeles Angels a day earlier.
Indians: Cleveland activated four players as part of their Sept. 1 roster moves: RHP Josh Tomlin (hamstring), OF Lonnie Chisenhall (calf) and OF Abraham Almonte (hamstring) came off the 10-day disabled list, while RHP Dan Otero was taken off the paternity list.
Tigers: Detroit activated LHP Daniel Norris (groin) and RHP Anibal Sanchez (hamstring) from the disabled list before the game. Both struggled in rehab starts on Thursday, and manager Brad Ausmus doesn't know when they will return to Detroit's rotation. ... RHP Michael Fulmer will miss at least two starts due to recurring numbness in his pitching hand. He won't start again until after a visit with Dr. James Andrews.
WELCOME TO THE BIG LEAGUES
Tigers reliever Jairo Labourt made his major league debut in the eighth inning of the nightcap. Two of the three batters he retired — Greg Allen and Francisco Mejia — were both batting for the first time in their own big league debuts.
COMING FULL CIRCLE
Verlander's major league debut came in a Tigers-Indians doubleheader on July 4, 2005. Twelve years later, Detroit started the post-Verlander era with another doubleheader against Cleveland.
The long-time Tigers ace wasn't quite ready to end his days in Detroit, though. Ausmus said Verlander visited the clubhouse during the first game.
The Tigers and Indians play the third game of the four-game series on Saturday evening. Jordan Zimmermann (6-11, 6.08) will face Cleveland ace Corey Kluber (13-4, 2.63). Zimmermann is 0-3 with an 11.42 ERA in four career starts against the Indians, while Kluber is 6-7 with a 4.47 ERA in 20 starts against Detroit.