Christian Eriksen and Marcus Rashford scored the goals for the 20-time English champions, who proved too strong for Vincent Kompany's side over the course of the 90 minutes.
Man United had five players in their XI who were involved in the World Cup, with Tyrell Malacia, Casemiro, Eriksen, Bruno Fernandes and Rashford handed starts.
However, Diogo Dalot (injury) and Harry Maguire (illness) were absent, while Jadon Sancho is still not training with the club, as he undergoes an individual training programme.
Lisandro Martinez and Raphael Varane were also missing due to their involvement in the World Cup final, which led to Casemiro starting in the middle of the defence.
Burnley carved out the first half-chance in the third minute when Johann Berg Gudmundsson delivered a brilliant cross onto the head of Ashley Barnes, only for the forward to guide his header wide.
The offside flag was raised, though, so it would not have counted even if it had found a route past Martin Dubravka in the home side's goal.
Burnley's early confidence was not a surprise considering that they entered the match off the back of a four-game winning run, with Kompany's side sitting at the top of the Championship.
Man United had a golden chance to make the breakthrough in the eighth minute when Fernandes released Alejandro Garnacho inside the penalty box, but the youngster could not poke the ball past the onrushing Bailey Peacock-Farrell.
The Red Devils took the lead in the 27th minute of the match, with Wan-Bissaka doing brilliantly to steer a cross towards Eriksen, who fired home from close range.
Anthony Martial then placed not into the arms of the Burnley goalkeeper in the 30th minute, with Man United looking dangerous when given time and space in the final third of the field.
Manuel Benson fired just wide of Dubravka's post three minutes later, before the on-loan Man United goalkeeper had to get across to keep out a powerful strike from the same player.
The Red Devils escaped a very dangerous moment shortly after, as Dubravka failed to make contact on a high ball, which then bounced towards goal, but Wan-Bissaka made the clearance.
Dubravka had another nervy moment when he allowed the ball to drop behind him after taking his eye off it, but the 33-year-old was able to make the clearance.
Down the other end, Peacock-Farrell did brilliantly to keep out a powerful effort from Martial, and an eventful opening 45 minutes finished with Eriksen's goal separating the two sides.
McTominay scooped wide of the Burnley post early in the second period after brilliant work from both Fernandes and Rashford, before Benson fired past the post down the other end.
Man United had their second in the 57th minute, though, when Rashford carried the ball from his own half into a shooting position before firing the ball into the far corner of the net; it was a spectacular effort from the England international, who gave the goalkeeper no chance with an emphatic finish.
Burnley had a golden chance to halve the deficit in the 68th minute when Gudmundsson found Darko Churlinov with a brilliant pass, but the substitute missed the target, before Barnes fired just wide moments later.
Ian Maatsen had one deflected wide of the Man United goal in the 81st minute, before Dubravka got down to keep out an effort from Gudmundsson moments later.
Barnes then missed a huge chance after a mistake from Casemiro before Peacock-Farrell kept out Antony down the other end, with the two teams ultimately separated by two goals on the night.
The Red Devils, who have won the League Cup on five previous occasions, will be back in Premier League action against Nottingham Forest on December 27. Burnley, meanwhile, will be bidding to return to winning ways when they host Birmingham City in the Championship on the same night.