United had Jose Mourinho and Ander Herrera sent off and found former keeper Tom Heaton in inspired form on a fractious afternoon at Old Trafford.
Just as against Stoke, the Reds were totally dominant and registered 38 shots on goal but were thwarted by a combination of bad luck, superb goalkeeping and questionable refereeing.
We drew a blank for the third successive league game with the tenacious Clarets indebted to their in-form keeper and the woodwork which was rattled twice by Zlatan Ibrahimovic and Juan Mata.
The tone of what was to follow came inside a minute when Heaton held out of form Zlatan's strike from distance and then denied Mata after good link-up play between the Swede and Marcus Rashford.
Paul Pogba went close with a deflected effort while Burnley - looking to avoid a fourth away defeat of the season - struggled to impose themselves.
Heaton was on a one-man mission to deny his former club and Zlatan in particular and came to his side's rescue again with a series of superb saves shortly before the break.
Pogba found Ibrahimovic who was kept out by an excellent save from the sprawling stopper before Mata's low effort from 12 yards was tipped around the post.
Herrera picked out Jesse Lingard who had a header turned over the top and Ibra's side-footed finish was diverted behind by Ben Mee after excellent play from Pogba.
The Frenchman flashed inches over the crossbar before strong penalty appeals were turned down after Matteo Darmian went down under a challenge from substitute Jon Flanagan.
That saw a furious Jose sent to the stands but his side continued where we had left off after half time.
Ibrahimovic worked his way to the byline and cut the ball back for Mata but the impressive Heaton saved with his legs and also blocked Lingard's follow-up.
Pogba's cross hit the hand of Michael Keane- again the appeals were waved away- but United continued to lay siege to a Burnley goal that was living a charmed life.
Heaton's best save of the lot saw him somehow miraculously fist away Ibra's point blank range effort but, even when he wasn't called upon, we were denied by the woodwork.
Zlatan headed against the bar and Mata's low effort hit a post before our afternoon became even more frustrating when Herrera was sent off for a second booking when he slipped and clipped the heels of Dean Marney.
Burnley could have snatched an unlikely win when Sam Vokes flicked narrowly wide from Scott Arfield's cross but United kept coming in the closing stages.
Daley Blind, Luke Shaw, Pogba and substitute Wayne Rooney all went close to the elusive breakthrough but Sean Dyche's side survived five added minutes to secure their first away point of the campaign.
Overall team performance: 9/10
United Faithful Man of the Match: Paul Pogba