The hosts took a two-goal advantage into half-time thanks to goals from Will Hugill and George Walters, before Sean Etakulu added a third against the run of play midway through the second period.
In windy conditions at the County Ground in Lancashire, it was a competitive start and both sides had early opportunities. It was Luton who threatened first with Jake Burger, after he received the ball on the edge of the box and fired towards goal but an important block from Keelan Williams sent it behind for a corner.
Minutes later Burnley threatened from a corner of their own, as Dan Sassi rose highest at the back post to head just wide, a chance upon reflection he could have done better with.
Tyrelle Newton started the game well and led a fine attacking move for the Hatters after winning possession and driving forward with the ball.
He picked out Josh Allen with a good pass, and with a flick in behind, Oliver Lynch found himself with only goalkeeper Sam Waller to beat but his effort was tame and easy to gather.
The Hatters went even closer just before the half-hour mark as Ed McJannet rattled the crossbar from range, before the hosts made Dan Walder’s side pay moments after.
After threatening with a number of corners during the opening stages, the Clarets took the front through Hugill inside the six-yard box. Sassi was once again the target and Jack Leckie managed to keep it alive as the ball dropped for the Burnley number eight, who beat Jacob Cowler from close range.
Deep into added time, Walters added a second for the hosts after good work from Etaluku on the wing. He kept the ball in play, drove into the box and forced a good save from Cowler, but the Luton ‘keeper was unable to stop the rebound that was smashed home.
The young Hatters showed plenty of fighting spirit as the second half began, twice going close to pulling a goal back in the opening five minutes.
Having beaten the offside trap, Newton latched onto a long ball and ran through on goal but was denied by the outstretched leg on Waller. The following corner led to another opportunity as Adam Wedd towered above his man at the back post but once more, the Burnley number one was equal to it.
But after Cowler was called into action to deny Ratchford twice in quick succession, the hosts did secure a third goal through Etaluku, courtesy of a wicked deflection off Wedd.
The defender was on hand to help prevent a fourth goal late in the game as he produced a vital interception and Cowler followed up by clearing the impending danger.
Victors Burnley will now face either Fleetwood Town or Liverpool away in the fourth round of the competition, while the Hatters will look to build on their 5th place standing in the EFL Youth Alliance Division when they return to league action.
Burnley: Waller; Williams, Leckie, Sassi, McCullough; Etaluku, Bauress, Hugill, Ratchford (Couch 74); Coppack, Walters (Bryan 77).
Subs not used: Casper, Couch, James, Westley, Behan, Grant.
Goals: Hugill (31’), Walters (45+2), Etaluku (65’)
Yellows: Sassi, Williams, Etaluku
Town: Cowler; Pinnington (Odell 77), Bateson, Wedd, Heron; Corbit (C), Burger, Newton, McJannet; Lynch (Matthews-Lewis 65), Allen.
Subs not used: Bentley, Nelson, Moffat, Martucci, Hemlin.
Yellows: Wedd, Corbit