Ed McJannet scored twice in the opening 45 before Elliot Thorpe added a second-half brace, followed by a fifth goal from Josh Allen in the closing stages.
Town enjoyed plenty of possession in the opening stages and went close to an early goal through McJannet.
Allen danced away from two defenders with good footwork, before McJannet took over on the edge of the box, dragging his shot just wide of the target.
Thorpe went even closer after Dan Gould was called into action to deny the midfielder twice from close range after Josh Allen’s cross deflected kindly into his path.
Moments later, Town would take the lead through McJannet, who rose highest to head home Josh Williams’ cross from the right-hand-side.
The Republic of Ireland U19 international doubled his tally with 10 minutes left of the first half with a low finish from the edge of the box, following a swift counter-attack.
Dunstable had a chance to reduce the deficit on the stroke of half-time after Terrence Muchineripi accelerated towards the byline and crossed well but Jacob Cowler was able to gather Jack Percy’s tame header.
It was a chance they would rue after Thorpe added a third for the Hatters after the hour mark. The hard-working McJannet forced the home side into conceding possession and Thorpe fired low into the bottom corner with Gould unsighted.
TQ Addy was looking for a fourth shortly after as he drove past a handful of players to race into the box but his shot was deflected behind for a corner.
Relentless in their pursuit for a fourth goal, it was just minutes later that Thorpe added his second and Town’s fourth.
Allen and Conor Lawless combined with a neat one-two inside the area. The latter had the awareness to then pick out Thorpe who charged into the box and lifted his shot above the goalkeeper and into the top corner.
In the closing stages Town scored a fifth through Allen after he tapped home from close range to round off a good team performance at Creasey Park.
The Hatters will now face Leighton Town in the second round of the competition, with date and time to be confirmed.
Dunstable Town: Gould, Creary, Mead, McCrohan, Sonuga (C), Naylor, Sellers-West, Faulkner, Muchineripi, Mills (Gray 65'), Percy.
Subs not used: Trif, Baker, Partley, Stirling.
Luton Town: Cowler; Bateson, Pettit (C), Panter; Pinnington, Thorpe (Corbit 72'), Lawless, Williams; Addy, McJannet (Matthews-Lewis 85'), Allen.
Subs not used: Bentley, Nelson, Newton.
Goals: McJannet (22', 35'), Thorpe (66', 71'), Allen (83')
Attendance: 365