Di Maria started the game, but despite some probing passes and darting runs, he could not illuminate United's path to goal against Sean Dyche's promoted side.
Having seen his team beaten by Swansea City and held by Sunderland before being humiliated 4-0 by third-tier Milton Keynes Dons in the League Cup on Tuesday, new United manager Louis van Gaal has now gone four competitive games without recording a win.
"We were the stronger team, certainly in the second half," Van Gaal told BT Sport. "We created chances, but you have to score. That is where we have failed. The progress is there, but you have to win.
"A club like Manchester United has to win. We have two points from nine and that's disappointing."
Burnley, with no points to show for their first two efforts, were hoping to replicate their first-ever Premiership match five years ago, when they stunned United 1-0 at Turf Moor.
United's frustrations were compounded in the last five minutes when Spanish playmaker Juan Mata had to go off after appearing to injure his hamstring.
Di Maria came into the United team in central midfield alongside Darren Fletcher and witnessed Burnley twice go close to taking the lead inside the first five minutes.
Burnley's former United trainee Dave Jones curled a free-kick against the bar and visiting goalkeeper David de Gea then had to save alertly after Lukas Jutkiewicz seized on a miscued back-pass by Jonny Evans.
Di Maria produced his first moment of quality in a United shirt in the 15th minute with a raking 50-yard pass for Robin van Persie, who chested the ball down, only for Burnley goalkeeper Tom Heaton to parry his shot.
Dave Jones worked De Gea from 25 yards at the other end, while United felt they should have been awarded a penalty after Burnley right-back Kieran Tripper appeared to push Ashley Young over in the area shortly before half-time.
Van Persie threatened twice for United around the hour, seeing a volleyed cleared off the line by Dean Marney and heading over, before Di Maria was replaced by Anderson with 20 minutes to play after taking a kick to the calf.
Van Gaal also sent Danny Welbeck into the fray, but with newly appointed England captain Wayne Rooney heading wide and Burnley substitute Ashley Barnes surviving a penalty-box handball claim, the Dutchman had to content himself with a point.