Ryan Succop kicked six field goals, including one to force overtime in the final seconds and another from 31 yards to complete the Chiefs' (1-2) comeback against the winless Saints (0-3).
Succop's 43-yard kick with three seconds left in regulation completed a methodical comeback by the visitors after New Orleans had cashed in on a pair of Chiefs turnovers to go ahead 24-6 in the third quarter.
In storming back to win for the first time this season Kansas City needed only one touchdown, a blazing 91-yard run by Jamaal Charles, who finished with 233 yards rushing and 55 yards receiving.
The Chiefs also got a safety in the fourth quarter on Justin Houston's third sack of the game, a score which would prove crucial come the final moments of regulation.
Drew Brees passed for 240 yards and three touchdowns, but missed all six of his passes through the fourth quarter and overtime, when the Saints were unable to get a single first down.
After Charles' long TD had made it 24-13 in the third quarter, Stanford Routt's interception of Brees' underthrown pass for Devery Henderson near the Kansas City goal-line thwarted another Saints scoring chance.
Succop field goals of 34 and 38 yards early in the fourth quarter cut it to 24-19, setting up a wild finish in which the safety cut the deficit to three points and gave Kansas City the ball with 5:33 left.
Matt Cassel's spinning scramble and 11-yard pass across the field to Jon Baldwin on third-and-10 extended the drive, which also included a completion to Dwayne Bowe on fourth-and-5 before Succop's game-tying kick.
In one of several lowlights for the crew of replacement officials working the game, New Orleans was briefly ruled the winner on the field in overtime when running back Shaun Draughn lost the ball stretching for a first down and safety Roman Harper picked up the ball and ran to the end zone.
The fumble was overturned on video review, but the spot came up a half-yard short of a first down. Charles easily converted a fourth-down run to extend the winning drive.
The Saints had been in control at the start, marching 83 yards in six plays before Lance Moore made it 7-0 with a leaping, outstretched snag of Brees' nine-yard pass.
After that, however, no team got in the end zone for the rest of the half. Succop hit field goals of 25 and 45 yards, and New Orleans' Garrett Hartley hit from 40, and the Saints led 10-6 at half-time.
New Orleans extended their lead due to two turnovers in the third quarter. First came a fumble by Dexter McCluster, who hurt his shoulder while falling on his own after a short catch. The Saints were awarded the ball after a review, setting up Brees for a one-yard TD pass to tight end Jimmy Graham.
New Orleans then went ahead 24-6 on Brees' six-yard swing pass to fullback Jed Collins, capping a short drive set up by Greer's interception and 28-yard return to the Kansas City seven, before the Kansas City resurgence started.