It took the hosts just five minutes to break the deadlock through Marcus Rashford, and the match was effectively wrapped up before half time as Rashford added a second before Jesse Lingard got in on the act.
Anthony Martial scored a fourth in the second half as United got their defence of the trophy off to a winning start, although their perfect run of not conceding at home this season came to an end in stoppage time when Lloyd Dyer grabbed a late consolation.
Both teams made nine changes to their sides from the weekend, and it was the visitors who had the first sight of goal on the evening when Luke Varney drew a save from Sergio Romero inside three minutes.
Burton's bright start was brought to a sudden halt when United broke the deadlock moments later, though, as Lingard produced a clever lay-off to steer Michael Carrick's pass into the path of Rashford, who coolly lifted his finish over the onrushing keeper and into the net.
Lingard had a chance of his own just one minute later when he was played through on goal, but a sliding block from Tom Naylor denied the United midfielder.
The pressure from the hosts was relentless, though, and Martial saw one shot saved before United scored their second after just 17 minutes, with Rashford firing a pinpoint effort in off the post from 25 yards to double his own personal tally for the evening.
Jose Mourinho's side looked dangerous almost every time they came forward and Juan Mata forced another save from the busy Connor Ripley before turning provider as the Burton keeper denied Martial when one on one shortly afterwards.
Burton's big chance of the first half arrived moments later when Varney poked the ball into the path of Joe Mason, who saw his attempted chipped finish thwarted by Romero from point-blank range.
United were immediately back on the front foot, though, and were within the width of the woodwork away from a third goal just one minute later when Mata nutmegged one defender in the box before firing his subsequent effort against the foot of the post.
The hosts would not have to wait much longer before they well and truly put the game to bed, with Lingard extending their advantage with just under 10 minutes to go until half time when his 25-yard effort deflected off Ben Turner to leave Ripley helpless as it nestled into the bottom corner.
Having seen any hope of an upset disappear before half time, Burton could have been forgiven for coming out for the second half in damage limitation mode, but the visitors had the chance to pull a goal back within two minutes of the restart when Hope Akpan was picked out by Matt Palmer's cross, only to send his stooping header well off target.
United soon resumed control of proceedings, though, and Martial had a sighter moments before adding the fourth when he danced his way through the Burton defence only to be denied a brilliant solo goal by a good low stop from the keeper.
The French winger did get in on the act two minutes later, though, combining with Lingard and Rashford to race through on goal before this time tucking his finish beyond Ripley and into the bottom corner.
It was Martial who continued to look most likely to add to United's lead, coming close with one effort across the face of goal before rippling the side-netting with a free kick two minutes from time.
However, Burton got the last laugh with the final meaningful action of the game as Joel Pereira - who had earlier come on as a sub for Romero - spilled the ball at the feet of Dyer, who duly lashed his finish home.
The consolation had no bearing on the result - which had not been in doubt since United opened the scoring after five minutes - but it did see the Red Devils concede at Old Trafford for the first time this season having previously scored 17 goals without reply in front of their own fans.
United have now reached at least the fourth round of this competition in nine of the past 10 seasons and are looking to become the first side to retain the trophy since Sir Alex Ferguson's United side of 2009 and 2010 achieved the feat, whereas Burton have still never made it past the third round.