Goals from Juan Mata, Marcus Rashford and Mason Greenwood were enough to hand Ole Gunnar Solskjaer's side their first win of the season, booking a fourth-round trip to either Preston or Brighton, who play tomorrow.
United were made to work, however, with a resolute Luton side showing why they sit third in the Championship with two wins from two, having progressed to the third round with impressive victories over Norwich and Reading.
With one eye on their tricky league trip to Brighton on Saturday, Solskjaer made 10 changes to the side that lost to Crystal Palace in their top-flight opener, handing first starts to Dean Henderson and summer signing Donny van de Beek.
Though United started the brighter, Luton grew in confidence as the half progressed and were undoubtedly the side on top as the clock hit 43 minutes.
United capitalised on a rare Luton mistake on the stroke of half-time, however, as Tom Lockyer's panicked clearance ricocheted off the arms of Andrew Shinnie and into the path of Brandon Williams, who was tripped in the box by George Moncur.
Mata made no mistake with the resulting penalty, stroking home into the bottom right-hand corner to give the visitors a half-time lead.
United looked to set a quicker tempo in the second half but were again unable to carve open any clear-cut chances until Solskjaer called for Bruno Fernandes, Rashford and Greenwood with 10 minutes remaining.
After Henderson was called into action with a fine save from a close-range Lockyer header at the near post, substitutes Greenwood and Rashford linked up, with the latter slotting home to put the game beyond doubt.
Greenwood then added another in the dying seconds, calmly passing beyond James Shea in the Luton goal as United progressed to the next round.
Prior to the game, Solskjaer insisted it would take his side up to five games to reach peak condition after a delayed return to pre-season following their run to the Europa League semi-finals last term.
And while this was indeed no vintage display - from a side missing the likes of Paul Pogba and Anthony Martial - a place in the fourth round and some additional minutes in the legs will certainly be pleasing for the Norwegian.