After kick-off was delayed by 20 minutes to allow Cambridge supporters stuck in traffic to arrive, Tom Elliott missed a glorious chance to give the League Two side a shock lead inside the first minute when he hit the post with just David de Gea to beat.
From then on it was one-way traffic and the hosts broke the deadlock after 25 minutes when Marouane Fellaini headed across goal for Juan Mata to finish.
Marcos Rojo doubled the advantage in the 32nd minute when he headed home from Robin Van Persie’s chipped cross.
And substitute James Wilson capped off a comfortable win for Manchester United when he fired home from the edge of the box with 17 minutes remaining.
Next up for Louis van Gaal’s side is a trip to Preston North End, who beat Sheffield United 3-1.
The Red Devils are the favourites to win the competition this year and Van Gaal showed how seriously he is taking it by naming a strong starting line-up – worth over £200m – against League Two Cambridge.
But the star-studded hosts should have been behind inside the opening minute as Daley Blind’s slip in midfield put Elliott through on goal.
Unfortunately for the striker he could not take the chance as he slid his shot past De Gea but off the post.
That was to be the visitors’ best opening of the night but their Premier League hosts struggled initially to create much at the other end as they appeared determined to cross the ball into the box at every opportunity.
It eventually paid dividends in the 25th minute as Angel Di Maria chipped a cross to the back post for Fellaini to nod down to Mata, who finished off the underside of the bar.
Seven minutes later the lead was doubled as Fellaini’s shot was deflected into the path of Van Persie and he stabbed in a cross with the outside of his boot that Rojo headed in.
Ryan Donaldson dragged a shot wide late in the half after a rare Cambridge attack but the second half was completely dominated by the hosts.
Chris Dunn saved from Van Persie and Di Maria before substitutes Ander Herrera and Wilson combined for the third goal.
Herrera drove at the Cambridge defence and fed the ball forward for Wilson, who fired a shot past Dunn from the edge of the box.
Cameron McGeehan had a chance to snatch a consolation effort in added time, but his shot flew just wide of De Gea’s goal.