Paul McGuinness's side were too slick for the Clarets and had the game as good as won with barely a half hour played.
Skipper Davide Petrucci put the Reds in front in the 22nd minute and the advantage was doubled six minutes later when Joshua King found the net with an angled drive. William Keane grabbed United's third with an inch perfect lob over goalkeeper Danny MacDonald.
King claimed his second six minutes after the restart before Burnley defender Tom Anderson was dismissed for a second bookable offence.
Joe McKee scored in the 84th minute to give home fans something to cheer, but United had the last word when Keane netted with two minutes remaining.
United will now play Blackburn Rovers or Leyton Orient in the fifth round.
UNITED: Devlin, Massacci (Fornasier 75), Ajose, Wootton, McGinty, Tunnicliffe, Cole, Pogba, W Keane , Petrucci (Lingard 68), King Cofie 63). Subs not used: Johnstone, Hussain.
Source: Manchester Evening News