Tyrelle Newton had fired the Hatters into the lead with seven minutes left to play but Luke Pyman equalised in added time to send the match to penalties.
The hosts were clinical from the spot and after misses from Josh Allen and TQ Addy, Town’s cup journey was ended at the second round stage.
Professional Player Development coach Adrian Forbes made five changes from the first round win against Dunstable Town last month, and Zack Nelson, one of those brought into the fold, came closest to opening the scoring early on.
Ed McJannet’s free-kick was met by Nelson, whose header bounced off the post and was cleared by the hosts, before they had a golden chance of their own in the 28th minute.
Eddie Corbit was dispossessed and the ball fell to Jervon Campbell, who couldn’t find the target with only Jameson Horlick to beat.
Nelson was once more a threat in attack, forcing Connor Coulson into a smart stop from close range after McJannet threaded the ball through to him.
It was an end-to-end affair however and Leighton had two good chances to take the lead either side of the half-time. Both guilt-edge opportunities fell to striker Ashton Campbell, but he failed to find the target on either attempt.
As the weather worsened during the second half, Josh Neufville was introduced just after the hour mark, his first involvement in any competitive fixture since May last year when he suffered an injury on loan with Yeovil Town.
It could have been a dream return for the winger as he was played though on goal and about to pull the trigger before Joe Fitzgerald produced a vital block to deny him.
Corey Panter was denied minutes later with a point-blank save from Coulson as the Hatters continued their quest for an opener, but to no avail.
In the 83rd minute however, they would get their reward. Newton found the back of the net after great work from Neufville. He drove through midfield and picked out Panter in space, whose cross was diverted across goal by Allen and finished by Newton.
But in added time, there was a final twist in the tale as Pyman slotted into an empty net after Horlick misjudged a cross, sending the game to penalties.
Allen and TQ Addy failed to find the target from 12 yards, as the clincial Leighton progressed into the next round, with Ross Adams scoring the decisive spot-kick.
Leighton: Coulson; Bryant, Williams, Pyman, Fitzgerald, Morgan, Adams, Osborne (Kerr 54’), A. Campbell, Towell, J. Campbell (Newbury-Barr 70’)
Subs not used: Foulger, Heeps
Yellows: A. Campbell, Towell, Fitzgerald
Goals: Pyman (90+2')
Luton: Horlick; Pinnington, Corbit, Francis-Clarke, Panter (C); Lawless, Newton, McJannet; Nelson (Neufville 63’), Addy; Allen.
Subs not used: Cowler, Bateson, Matthews-Lewis, Luker.
Goals: Newton (83')