Luck needed only 74 yards to break the Carolina quarterback's mark of 4,051 yards, and he eclipsed it while leading the Colts to a field goal that gave Indianapolis a 13-3 halftime lead.
Luck threw for one touchdown and a total of 205 yards, giving him 4,183 on the season.
In addition to Luck's feat, the Colts could also celebrate a trip to the postseason after sending off the Chiefs.
Luck had to again rally Indianapolis for the win, giving him a rookie quarterback record of seven comebacks in the fourth quarter. His 21 touchdowns trail Russell Wilson's 25 as the most for a rookie quarterback this season. Both are behind Peyton Manning's record of 26 set in 1998.
Information from The Associated Press was used in this report.