First half goals from Reece Burke and Carlos Mendes Gomes, both their first for the club, and a late one from Admiral Muskwe ensured a smooth passage through a tricky tie at the Abbey.
Manager Nathan Jones made eight changes from the side that won 1-0 at Swansea in midweek.
Jed Steer was picked for his Luton debut between the sticks, and Elliot Thorpe started in midfield, his first involvement in the matchday squad and a career senior debut for the ex-Tottenham youngster.
Reece Burke, Gabe Osho and Fred Onyedinma keep their places from Tuesday's away win against Swansea City, whilst there is a start for Carlos Mendes Gomes in a forward role.
Exciting youngster Josh Neufville, 20, was making his return from a broken ankle suffered last May while on loan at Yeovil and was on the bench, as were development squad regulars Ed McJannet, 18, and 17-year-old Aidan Francis-Clarke.
With a full house behind them, Cambridge made the early running, Wes Hoolahan and Adam May both firing over in the opening ten minutes before Jerome had the Town's first sight of goal, but his attempt at steering Muskwe's low cross in was wide of the near post.
It didn't take long for the Hatters to break the deadlock, with Burke meeting Thorpe's right-wing free-kick with a header into the bottom corner, after Muskwe had been fouled by Harrison Dunk.
That was in the 14th minute, and by the 23rd it was two, Jerome doing well down the left channel and squaring the ball across the edge of the Cambridge box for Mendes Gomes, who evaded the lunging challenge of Dunk to slam his shot past Dimitar Mitov to open his Town account.
It was plain sailing for the remainder of the half, with the hosts threatening briefly and having a penalty appeal turned down as a ball ricocheted towards Peter Kioso's arm, but referee Joshua Smith was unmoved and Town remained well in control.
Tom Lockyer should have extended the advantage ten minutes after the break, instead firing over from close range after Muskwe had nodded Gabe Osho's cross into his path.
Cambridge spurned a chance to get back into it on the hour when Sam Smith dragged a shot wide from the edge of the Town box, then Wes Hoolahan fashioned a shooting opportunity, but didn't pull the trigger and Steer remained untroubled.
Jones replaced Thorpe with Danny Hylton with just over 20 minutes to go, then after Cambridge boss Mark Bonner had made a triple substitution, Jones brought midweek winning goalscorer Harry Cornick on for Jerome with just under quarter of an hour left.
Steer had to be alert to fall on an 80th-minute cross from Hoolahan that looked to be arcing inside the far post, then he got down low to his left to push away a 20-yarder from sub Ben Worman, with Smith rattling the crossbar with the follow-up, although it wouldn't have counted with the linesman's flag raised for offside.
The Hatters wrapped it up in the 89th minute with Muske bending one in from the right angle of the box, seconds after sub Campbell had seen a shot blocked, with the aid of a deflection off Sam Sherring.
Town will be ball number five in tomorrow morning's fifth round draw, due to take place at 11.40am in the build-up to the Liverpool v Cardiff tie on ITV.
Goals:
Luton - Burke 14, Mendes Gomes 23, Muskwe 88
Att: 7, 937 (1,470 away)
