Peyton Manning had a pedestrian day, completing 26 of 44 passes for 244 yards with an interception for the Colts (3-2), who rebounded from a last-second loss at Jacksonville in Week 4.
"Mike did a nice job for us. That was a heck of an effort to get the ball in the end zone. He did some good things for us," Colts head coach Jim Caldwell said of Hart, who saw extra time after Addai went down with a shoulder injury in the third quarter.
Adam Vinatieri nailed four field goals, while his Chiefs counterpart, Ryan Succop, went 3-for-4, missing a 51-yarder in the final two minutes.
Matt Cassel threw for a mere 156 yards on 16 of 29 passing for the Chiefs (3-1), who were trying for their first 4-0 start since 2003. Jamaal Charles logged 87 yards on 16 carries and lost a fumble in the setback.
The Colts were the last unbeaten team to fall in 2009.