The hosts handed a recall to Marcus Rashford in place of Alexis Sanchez, while Andreas Pereira was given a start in midfield, with the Clarets naming an unchanged XI from their last league outing.
And it was Rashford who passed up a fine early chance for the hosts as he exchanged passes with Romelu Lukaku and raced clear, only to place his shot well wide of target as Tom Heaton came off his line.
Just over 20 minutes in Rashford was the provider, but even as Juan Mata saw his shot strike the outside of the post, the offside flag had already been raised.
Shortly before half-time the hosts did have the ball in the net, with Mata this time the provider for Paul Pogba to slide the ball past Heaton, but again an offside flag cut short the celebrations before they had started.
Mata drew a save from Heaton early in the second period, but the effort was too weak to cause the Clarets keeper any real problems.
And six minutes into the second half it was the visitors who broke the deadlock, as Jack Cork took the ball off Pereira and fed Ashley Barnes, who slammed his finish past David de Gea to silence most of Old Trafford.
The hosts responded strongly and introduced Jesse Lingard in an attempt to turn the game around, with Heaton making a superb save to turn aside a Lukaku effort and preserve the Clarets' lead.
However, United were caught by a sucker punch nine minutes from time as Chris Wood doubled the Burnley lead, heading home from an Ashley Westwood cross.
Three minutes from time United were thrown a lifeline as Jeff Hendrick fouled Lingard in the area and Pogba made no mistake with the resulting spot-kick.
Pogba was then denied by Heaton as the hosts poured forward looking for an equaliser and they struck deep into stoppage time to salvage a point, as Victor Lindelof turned the ball home after Heaton did superbly to keep out a Sanchez header.