Luck scored on first-half runs of 3 and 5 yards while going 16-of-29 through the air for Colts (3-3), who barely outpaced the Browns 321-319 in total yardage.
Reggie Wayne led the Colts' receiving corps with 73 yards on six catches while Vick Ballard paced the runners with 84 yards on 20 carries in the victory.
Cleveland (1-6) couldn't eke out a second straight win despite Brandon Weeden's 264-yard, two-touchdown performance.
Josh Gordon notched a receiving touchdown en route to 59 yards for the Browns.
Celebrity Families of 2014 [PHOTOS]