After a thrilling but goalless first half, United found themselves a goal down through an unfortunate Lisandro Martinez own goal after 63 minutes.
The Cottagers looked like securing all three points until a deflected Matheus Cunha leveller with 89 minutes played.
The result saw Michael Carrick's team end the weekend in 12th place with five points from their opening five games of the new Premier League season.
Alvaro Arbeloa's team are stuck in 19th position with just two points.
The home side got on the ball first, and a good Antonee Robinson pass picked out Gonzalo Garcia on the edge of the area, but he could only drag his shot wide of the target.
United looked dangerous for the first time after 10 minutes as Youri Tielemans whipped a decent ball to the front post, where Bruno Fernandes won a corner from his deflected header.
Martinez then forced Bernd Leno into an excellent two-handed save with a veritable piledriver off the outside of his boot, before Garcia was a toe away from handing Fulham an 18th-minute lead as he looked to connect with a good cross from Alex Iwobi at the other end.
Josh King wasn't too far off with a fierce hit from just wide of the target after being fed in by Oscar Bobb.
Marcus Rashford headed over at the back post as Bryan Mbeumo broke forward and crossed from the left, while Diogo Dalot was wide with a curling effort from outside the 18-yard area as the away team looked to take control of the tie.
Bruno Fernandes thought he had scored with a 25th-minute shot that came back off the upright. The skipper was fed a perfect cut-back by Cunha after he expertly outmuscled David Affengruber before picking out his captain.
Senne Lammens then needed all of his reach to keep out a strong, low shot from Sander Berge before Bobb fired just wide after a superb turn inside the area.
Chances kept coming, and Cunha hit the frame of the Fulham goal after a huge deflection took the ball away from goalkeeper Leno. The resulting corner only led to a swift Cottagers counter-attack and a superb goalscoring chance, followed by an even better save, as Lammens kept out a strong King shot with his outstretched right boot.
Rashford lacked the power to beat Leno with his effort on the 40-minute mark, in what was the 23rd attempt at goal — the most of any Premier League game this season without a goal being scored. The half ultimately ended with 26 chances.
The very next effort saw Lammens quick off his line to dive at the feet of Iwobi as he looked to get on the end of a clever Garcia pass into the area, while Rashford sidefooted over the target when it might have been easier to score with the last real effort of an action-packed first 45 minutes.
There were no changes at the start of the second half, but the tempo of both teams was noticeably lower.
It took until the 53rd minute for a superb Tielemans pass to release Rashford down the left flank, but the England attacker chose to square the ball to Mbeumo, and Robinson was allowed to nip in and end the danger.
With 10 minutes of the second half played, Reds boss Carrick sent his substitutes out to warm up.
With the hour mark ticking over, Iwobi saw his goal-bound shot deflected wide by Dalot, but Fulham were soon ahead as a Calvin Bassey shot hit the knees of Lammens before rebounding off Martinez and into the back of the net.
The United goalkeeper may well have used his arms to make the save but instead allowed the ball to hit his body after making so many similar saves earlier in the match.
Leno kept Fulham's lead intact with an exceptional one-handed save to stop Mbeumo from curling home on 67 minutes.
Cesar Palacios and Kevin came on for Iwobi and Bobb for Fulham as they looked to cement their advantage with some fresh legs.
Leno made another strong save, this time to deny Cunha as he whipped in a shot seemingly from nowhere with two defenders at his hip.
Lammens then parried a good Kevin shot to safety, ahead of another Fulham change, with Hugo Larsson coming on for Berge, as Joshua Zirkzee replaced Rashford for the chasing team.
Fulham continued to ring the changes, with Rodrigo Muniz and Joachim Andersen coming on for Garcia and King, while Mason Mount came on for Kobbie Mainoo for United.
Carrick's team were seeing all of the ball in the dying moments, with Fulham happy to sit back and grind out the three points.
A last throw of the dice for United saw Patrick Dorgu introduced for Luke Shaw, with five minutes of the match to play.
Affengruber made an important header to stop Dorgu from getting on the end of a Mbeumo delivery before a quick VAR check confirmed there wasn't a handball in the Fulham penalty area from the subsequent corner.
But just when it appeared as if Manchester United were heading for a third defeat in a row in all competitions, they found an unlikely equaliser via a massively deflected Cunha shot, with 89 minutes on the big screen, ahead of a three-week break for international football.