The Colts are now level with the Houston Texans and Tennessee Titans on 6-6 in the tightest division in the AFC.
Luck, returning as starter after missing the Colts' previous game following a concussion, picked out tight end Dwayne Allen for three first-half touchdowns to set up a win which leaves Indianapolis tied for first in the AFC South.
The Colts are now level with the Houston Texans and Tennessee Titans on 6-6 in the tightest division in the AFC. The Colts host Houston on Sunday aiming to edge clear in the division.
Luck meanwhile showed no lingering after-effects from his concussion, going 22 of 28 for 278 yards. His fourth touchdown pass was a three-yard lob to Donte Moncrief in the third quarter.
Running back Robert Turbin barreled over from five yards for the Colts' fifth score of the night six seconds into the fourth quarter.
That was the cue for Luck to be removed from the fray for understudy Scott Tolzien.
Allen meanwhile was left celebrating a feast of touchdowns. His three six-pointers were equal to his total from his previous 22 games stretching back to the start of the 2015 season.
But the defeat sealed another miserable season for the Jets, who fell to 3-9 and are now eliminated from playoff contention.
The game may well have marked the end of an era for quarterback Ryan Fitzpatrick, who was benched after the second quarter in favor of Bryce Petty.
Petty went 11 of 25 for 135 yards with one touchdown, an impressive 41-yard connection with Robby Anderson.
Fitzpatrick, 34, made just five of 12 attempts for 81 yards and an interception in a poor first half display. The veteran becomes a free agent at the end of the season.