Nugent had already nailed a 33-yarder, but had it negated on a holding call. After a timeout, he came back to put the 43-yarder through the uprights and give the Jets the lead.
Kellen Clemens threw for 115 yards and a touchdown on 13 for 25 passing for the Jets (4-12), who ended their season by snapping a three-game losing streak.
Thomas Jones rushed for 98 yards on 25 carries and caught a touchdown pass in the win.
Brodie Croyle completed just 20 of 43 pass attempts for 195 yards and a touchdown for the Chiefs (4-12), who wrapped up the season on a nine-game losing streak.
Tony Gonzalez caught seven passes for 64 yards, breaking Shannon Sharpe's NFL record of 815 receptions by a tight end. Gonzalez now has 820 catches.