The Argentina midfielder, signed for a British-record fee of 59.7 million pounds ($99 million) from Real Madrid on Tuesday, played in central midfield and was one of United's best players at Turf Moor before being substituted in the 70th minute.
Di Maria set up United's best two chances of the first half, his raking ball setting up Robin van Persie for a chance that was parried away by goalkeeper Tom Heaton while Juan Mata slipped as he attempted to convert Di Maria's cross.
United has just two points from three games under new manager Louis van Gaal despite spending about 130 million pounds ($215 million) this summer. That figure is likely to grow before Monday's transfer deadline, with United announcing hours before the game that it has reached an agreement with Ajax for the signing of versatile Netherlands international Daley Blind.
Sticking with his 3-5-2 formation after Tuesday's embarrassing 4-0 loss to third-tier MK Dons in the League Cup, Van Gaal will be happy to see his much-maligned defense keep a first clean sheet but there was a lack of cutting edge going forward. That despite fielding Di Maria alongside Mata, Van Persie and Wayne Rooney in an exciting-looking forward line.
Burnley earned a first point since promotion from the League Championship and was comfortable for most of the game, although Di Maria stretched them at times.
He displayed a good range of passing _ exemplified by that brilliant long ball for Van Persie's chance in the 15th _ and always seemed free in the middle of the pitch. He also showed a good turn of pace to spark attacks, but faded after his impressive start. After flexing his legs following a heavy challenge, he was replaced by Anderson.
Van Persie had a goalbound shot cleared away from near the line in the 59th and the Netherlands striker looked United's most dangerous forward on a day when Rooney and Mata, who went off with a right hamstring injury, failed to have an influence.
Burnley came close early when David Jones curled a free kick against the crossbar, but the hosts mostly defended after that.