United were the better side for large parts of the game and controlled the tempo but were denied by goalkeeper Marc-Andre ter Stegen who made three outstanding saves in the first half.
LaLiga leaders Barcelona struck first, five minutes after the break, when Marcos Alonso scored with a towering header at the far post from a corner.
Marcus Rashford levelled three minutes later following a counter-attack down the right, slotting home a smart low finish between Ter Stegen and the near post.
The in-form Rashford also created United’s second, sprinting into the box from a short corner and delivering a low cross into a crowded six-yard area where the ball deflected into the goal off defender Jules Kounde.
United players and manager Erik ten Hag complained strongly after what looked like a clear foul by last-man Kounde on Rashford just outside the box, an infringement that should have produced a red card for the French international.
“(When) you face the number one (team) of Spain, and in an away game and you are that dominant, you can see that as a positive, yeah.” said Erik ten Hag
“Yes, we should have won this game. So, we have steps to go, but I think they were two good teams at (the) highest level, it was enjoyable to watch. But I think everyone agrees that it was a really high performance level from both teams, and when I show my team that I was very happy with our performance to face this strong team.”
“Yeah, I'm a little bit disappointed that we didn’t score them and I think we should have score minimum four goals tonight, and also I have to make the point, as a big influence not only on this game but around, it was at 2-1 a clear fault on Marcus Rashford.”
Barcelona dug deep to grab the equaliser, however, and mistakes by United's defence gifted the ball on the edge of their box to Raphinha whose cross eluded David de Gea and went into the net.
Barca made a late push for the win and could have scored when an Ansu Fati close range strike was denied and Casemiro's mis-hit clearance came off the crossbar.
"I have the feeling of being very satisfied despite the draw, even though we are Barca and we never like to draw. But I have the feeling that we competed against a rival that right now has a spectacular level in Europe, spectacular. I think it's one of the best teams in Europe right now, as a block, as a team. They are well-trained by a great coach and we have competed. We put our best foot forward. The attitude was immense. We matched them in intensity. We dominated in many moments, and I'm left thinking about the final push we made at the end when we showed lots of character." said Xavi
"On paper, (it's going to be) a game with a lot of intensity at Old Trafford. Lots of passion there at a spectacular ground. But I think that we can win. I had the feeling that we were able to win today in many moments of the game. This is the feeling that we are left with. We can go to Old Trafford to compete, and we'll see what happens."
"How are you going to feel? They didn't give us (a penalty for a hand ball in the box) in Milan in the Champions League, and we didn't get one today either. So, I don't know. I don't know what has to happen for a referee to blow his whistle for a penalty due to hand ball. It was very obvious, very obvious hands were used. They even went back and checked it, and the fourth (official) told me that they checked it and they said no. Well, I think it's incredible. Incredible."
Barcelona will travel to next week's second leg at Old Trafford with a depleted midfield as Pedri came off in the first half with a leg injury and Gavi will be suspended.