The visitors took the lead when Josh Windass burst down the right wing, before crossing for Scott Brown to turn in from close range.
Luton were unable to respond, and saw efforts from Windass and Terry Gornell go narrowly wide as they struggled to keep out the visitors.
Accrington sealed the win when Gornell met Tariqe Fosu's excellent cross with a glancing header.
Accrington rise into the automatic promotion places, two points clear of the play-offs with five games to go.
Accrington made it seven games unbeaten as they brushed aside Luton.