The 20-time English champions, who defeated Vincent Kompany's team in the whole 90 minutes with to goals from Christian Eriksen and Marcus Rashford, won the match.
Five World Cup participants made up Man United's starting lineup: Rashford, Casemiro, Eriksen, Bruno Fernandes, and Tyrell Malacia.
Diogo Dalot (injury) and Harry Maguire (sickness) weren't there, though, and Jadon Sancho still doesn't practise with the team since he has an individual training schedule.
Due to the absence of Lisandro Martinez and Raphael Varane as well as their participation in the World Cup final, Casemiro started in the middle of the defence.
In the third minute, Burnley created the first half-chance when Ashley Barnes misdirected his header after Johann Berg Gudmundsson produced a beautiful cross into the box.
Even if it had managed to go past Martin Dubravka in the goal for the visiting team, the offside flag had already been raised, therefore it would not have counted.
Given that Burnley came into the game with a four-game winning streak and Kompany's team leading the Championship, their early confidence was not a surprise.
When Alejandro Garnacho was sent free by Fernandes inside the penalty area in the eighth minute, Man United had a wonderful opportunity to score, but the youngster was unable to push the ball past the advancing Bailey Peacock-Farrell.
In the 27th minute of the game, Wan-Bissaka did an outstanding job of steering a cross toward Eriksen, who scored from close range.
In the 30th minute, Anthony Martial fired a shot that was saved by the Burnley goalie as Man United looked threatening in the final third of the field.
Three minutes later, Manuel Benson missed the post with a shot just wide. The on-loan Man United goalkeeper then had to sprint across the goal to block a strong shot from the same man.
Shortly after, the Red Devils narrowly avoided a potentially hazardous situation when Dubravka failed to make touch with a high ball that rebounded toward goal but was stopped by Wan-Bissaka.
Dubravka experienced another uneasy moment when he lost focus and let the ball to drop behind him, but the 33-year-old was still able to make the clearing.
An exciting first 45 minutes came to a close with Eriksen's goal separating the two teams after Peacock-Farrell made an outstanding save to save a thunderous drive from Martial down the other end.
After excellent work from both Fernandes and Rashford early in the second stanza, McTominay scooped wide of the Burnley post before Benson shot past the post down the other end.
But in the 57th minute, Rashford delivered Man United their second goal, advancing the ball from his own half into a shooting position and burying it into the far corner of the goal with a magnificent effort that left the goalie with no chance.
When Gudmundsson's superb ball to Darko Churlinov in the 68th minute was intercepted, Burnley had a wonderful opportunity to reduce the score. However, the replacement missed the goal, and minutes later, Barnes just missed the target.
In the 81st minute, a shot from Ian Maatsen was deflected beyond of the Man United goal, and seconds later, Dubravka dived to block a shot from Gudmundsson.
After Casemiro erred, Barnes had a golden opportunity to score but squandered it. Peacock-Farrell then denied Antony at the other end of the field, and the two teams finished the game separated by two goals.
The Red Devils, who have won the League Cup five times, will return to Premier League play on December 27 when they take on Nottingham Forest.