Fulham went in front in the 63rd minute when Calvin Bassey's low drive was parried by Senne Lammens and the rebound struck Martínez and trickled into the net. United created the better chances before the break, with Bruno Fernandes hitting a post from Cunha's cutback and Bernd Leno making saves to deny Bryan Mbeumo and Cunha, but the visitors did not level until the 89th minute, when Cunha took Patrick Dorgu's pass and shot from the edge of the box, the ball deflecting off David Affengruber and wrong-footing Leno.
The draw leaves Fulham without a league win after five games under Álvaro Arbeloa, though it is a second successive draw after the goalless one against Liverpool. United have one win, two draws and two defeats.
Arbeloa told BBC Match of the Day:
“I'm upset because we were so close, very unlucky with the goal we conceded. It is what it is, we need to keep working and improving. We knew this month would be difficult against top teams. I think we are in a good way.
“Both goalkeepers did good saves, many chances for both teams, two strange goals. Obviously we are upset to concede at the end.
“Calvin [Bassey] is one of the best at the back in the Premier League, he plays at a high level with and without the ball. He is doing so good.
“We want points, that's the most important thing, but I think my players are working hard and open to learn. They are playing in the way we want to play. If we continue playing this way the victories will come.”
Michael Carrick, speaking to BBC Sport, said:
“I think we did enough to win the game. I think we could win it. It was an unfortunate own goal but for the most part we were good. Their goalkeeper has made top saves, on another day we could have two or three goals. I was pleased with how we responded, the boys kept their belief and kept doing the right things. I think a point is the least we should take.”
On Fulham's goal, the Manchester United manager added:
“I think it was maybe a foul on Matheus [Cunha] first, there's an argument for that. The shot has come through someone and it has happened quite quickly. Quite a few moments like that have been going against us. Thankfully we have still taken something small from the game.”
United host Tottenham Hotspur on 10 October, the day Fulham visit Ipswich Town.