Cleveland traded for Harrison from Jacksonville before the start of the season, and he got his first start with the Browns on Sunday after a hamstring injury sidelined Karl Joseph.
Harrison's pick-six was just the Browns' second defensive touchdown in the last three seasons, and Rivers' 25th career interception returned for a score.
Colts rookie Isaiah Rodgers helped soften the blow when he returned a kickoff 101 yards for a touchdown.
Rodgers started along the right side where he toed the sideline in front of his teammates before he stopped to elude a Cleveland defender near midfield and cut back to the left, where he ran the rest of the way untouched into the end zone.
That was the Colts' first kickoff return for a touchdown since Jordan Todman returned the opening kickoff 99 yards for a touchdown against Green Bay in Week 9 of the 2016 season.
Per NFL Next Gen Stats, Rodgers ran 135.4 total yards on the kickoff return TD, the most total yards run by a ball carrier on any play this season.