Fulmer (9-6) gave up three runs, two earned, on seven hits with one strikeout and one walk in six-plus innings. Justin Wilson came on in the eighth inning to get the last five outs and pick up his 10th save.
Cleveland's Nick Goody (1-1), in relief of starter Corey Kluber, gave up three runs in one-third of an inning to take the loss.
Kluber was removed after five innings, mostly because his pitch count was at 101. He gave up one run on three hits with eight strikeouts and three walks.
Goody came in to start the sixth inning with the score tied at 1-1. Miguel Cabrera led off with a walk and was forced out at second on a groundball hit by J.D. Martinez. Victor Martinez singled, moving the runner to second, and Nicholas Castellanos walked to load the bases.
Dan Otero relieved Goody, and Alex Presley greeted Otero with a double to left field, scoring J.D. Martinez and Victor Martinez, giving Detroit a 3-1 lead. Castellanos went to third on Presley's hit, then scored on an infield single by Iglesias to make it 4-1.
An RBI single by Castellanos in the seventh stretched the lead to 5-1.
Fulmer held Cleveland to one run on three hits through six innings before struggling in the seventh. Edwin Encarnacion led off with a single, and he rode home on Jose Ramirez's 17th home run, a shot into the right-field seats that cut Cleveland's deficit to 5-3.
Fulmer gave up singles to the next two batters and was removed from the game. Shane Greene escaped the jam, then got the first out in the eighth before Wilson entered.
Kluber retired the first eight batters he faced before giving up a two-out double to Iglesias in the third inning. Ian Kinsler followed with a soft single to center field on an 0-2 pitch to score Iglesias with the first run of the game.
The Indians scored an unearned run in the fourth inning to tie it. Bradley Zimmer led off with a single. Fulmer threw wildly to first on a pickoff attempt, allowing Zimmer to advance to second.
NOTES: Indians OF Lonnie Chisenhall left in the fourth inning due to a right calf injury. ... Cleveland 2B Jason Kipnis was placed on the 10-day disabled list with a right hamstring strain. To replace Kipnis on the roster, the Indians recalled 3B Giovanny Urshela from Triple-A Columbus. ... Tigers 1B Miguel Cabrera was 0-for-1 against RHP Corey Kluber, but Cabrera's career batting average against Kluber is still .434 (23-for-53) with three doubles, six home runs and 12 RBIs. ... Detroit RHP Shane Greene has inherited 34 baserunners, and only four have scored. He entered with two on and no outs in the seventh inning Sunday and didn't let a runner come home. Greene's 12.5 percent inherited-runners-scored percentage leads the American League.