Louis van Gaal is yet to win a competitive game as a United manager, as Di Maria made his debut in Turf Moor, but was barely effective.
David Jones almost gave the hosts an early lead in the third-mimute when he tried a curler from a 25-yard freekick, but his autodidact effort hit the crossbar with David DeGea hapless in goal.
Sooner, the away side got in the mood, but a Robin Vanpersie shot in the 15-minutes was saved by Burnely’s goalkeeper Tom Heaton.
Di Maria sent in a mouth-watering pass to tee the Dutch striker, but his effort was blocked by Dean Marney.
United’s appeal for a penalty was denied in the closing stages of the second half, when Ashley Young’s shot met Ashley Barnes.
New signing Di Maria was substitute with 20 minutes to go, with what looks like a kick to the calf.
Burnley manager Sean Dyche, who was full of praise for his team’s display said afterwards:
“I thought we were productive in the first half. In the second half, we started very brightly and as the game wore on they began to control it better and we just got on the back foot a little bit.
“Overall we are happy with a point. We can’t be greedy, they are a good side”.
“I don’t think there is a lack or organization to the team, It’s the quality in the final third that wins you games and we were close again today.”