United dominated play early and scored a goal canceled by the referee due to offside through the video assistant referee, before Paul Pogba scored, in his first appearance in the starting lineup since mid-October, his first goal of the season, to give the visiting team a deserved advance.
United created plenty of chances to consolidate their lead before the break, but Burnley goalkeeper Nick Pope made it clear, while the home side did not hit a shot on target in the first half.
United paid the price for missed opportunities, but, contrary to the course of play, Jay Rodriguez equalized after a quick counter-attack in the 47th minute.
United could not repeat their form in the first half, and was frustrated after conceding the goal, but tried to press towards the end without success.
United fell to fifth place in the league with 39 points from 23 games, one point behind West Ham United, fourth.
With Newcastle United winning against Everton on Tuesday also, Burnley have fallen further behind in the relegation race as they are four points behind Eddie Howe’s side in 17th place, but Burnley have two more games to play.
Despite the absence of veteran striker Cristiano Ronaldo and sitting on the bench at the beginning, United’s attack was dangerous during the first half.
United defender Raphael Varane thought he scored his first goal for United after 12 minutes with a header, but the goal was canceled by the referee after a long review of the video referee for offside Harry Maguire.
United advanced well after that, as Marcus Rashford passed the ball to his colleague, who rushed from behind, Luke Shaw on the left side, and sent a cross, which Pogba met with a powerful shot into the net.
United continued the pressure and scored another goal that was canceled by the referee due to a foul against Rashford, before Edinson Cavani approached the goal with a header had it not been for the brilliance of goalkeeper Bob.
Rashford hit a powerful ball, which Bob saved again amid continuous pressure from United, but without succeeding in adding the second goal, to end the first half after the visiting team made 12 attempts on goal.
Rodriguez took advantage of a rare opportunity to score his first goal with Burnley this season, and then his fellow striker, newly joined Fout Weghorst, came close to achieving a surprise, as he almost scored for the host with a long-range header, but goalkeeper David de Gea saved the attempt brilliantly.
Varane almost scored with the heel of the foot, but United came out with a draw, and the team was satisfied with two wins in the last five games.