After a goalless first half, Calvin Bassey scored his first goal for the Cottagers midway through the second half, before Harry Maguire seemed to have rescued a point for the Red Devils with a close-range finish in the 89th minute.
However, more drama unfolded in stoppage time as Fulham burst down the other end to score a dramatic 97th-minute winner when Alex Iwobi nestled his strike into the bottom corner.
Man United's perfect run of results in February is now over and defeat for Erik ten Hag's side comes as a huge blow regarding their top-four hopes, as they remain sixth and have slipped eight points behind fourth-placed Aton Villa, who beat Nottingham Forest 4-2 on home soil.
As for Fulham, they celebrated their first win at Old Trafford since 2003 and have ended their 11-game winless run on the road, although they remain 12th in the Premier League table, three points behind the top 10.
Both teams made a cagey start to the contest, but it was Fulham who initially looked the livelier of the two sides in possession, quickly moving the ball through the lines, and former Red Devils man Andreas Pereira tried his luck with the visitors' first attempt on goal just three minutes in, one that was comfortably saved by Andre Onana.
Iwobi was then presented with two decent chances within the space of four minutes, but the Cottagers midfielder dragged both his efforts from the edge of the penalty area wide of Onana's right-hand post.
Just after the 20-minute mark, Man United's No.24 made a crucial one-handed save to keep out a glancing header from Fulham's Rodrigo Muniz following a corner, before Sasa Lukic failed to bundle home the rebound from close range.
The Red Devils had their fair share of possession but they were unable to create any clear-cut chances until a mix-up in the Fulham defence occurred in the 26th minute when defending a set-piece, allowing Alejandro Garnacho to pounce and curl a shot from inside the area.
However, Antonee Robinson got back well to head the ball off the line, before Maguire, who stated forward, hooked the rebound over the crossbar.
Both teams were then denied by the woodwork shortly after the half-hour mark; Diogo Dalot firstly saw a long-range strike clip the base of Bernd Leno's right-hand post, before Muniz at the other end used his strength to turn Lindelof inside the area before smacking the outside of the post with a ferocious strike.
The closely-fought contest became stretched as the first half came to a close; Onana made a fine save at full stretch to keep out a curling strike from Pereira after Lukic caught Casemiro in possession on the counter-attack. Leno then produced an almost carbon copy stop at the other end after launching himself into a one-handed save to keep out Garnacho's curler heading for the bottom corner.
Moments after the restart, a nasty clash heads during an aerial duel between Casemiro and Harrison Reed forced the medical teams of both clubs to enter the field; while the latter was deemed fit to continue, the former was replaced by Scott McTominay - Christian Eriksen was also introduced for the hosts at the expense of Forson, who looked lively in spells on his first senior start.
The first big chance of the second half fell to Fulham in the 56th minute, when Muniz dispossessed Eriksen on the halfway line and drove forward before teeing up Robinson, who opted against shooting from a tight angle and played a dangerous delivery across the six-yard box, begging to be tapped home, but Wilson was unable to meet cross.
United struggled to get a foothold of the match and groans reverberated around Old Trafford as Fulham capitalised on the hosts' sloppiness in possession to take control of proceedings - Marco Silva's side just lacked an end product in the final third.
Onana had to be alert to stretch and tip Reed's mis-placed cross from the right flank over the crossbar, but from the resulting corner, the ball fell kindly to Bassey, who smash a shot with his weaker right foot into the roof of the net to put Fulham into a deserved lead on the 65-minute mark.
Man United responded well as Fernandes weaved his way into a shotting position on the edge of the box, but his strike was comfortably collected by Leno to his left, before the Fulham shot-stopper got fingertips to Rashford's inviting delivery from an acute angle on the left wing.
Although Silva switched to a back following the introduction of Issa Diop, Fulham showed no initial signs of parking the bus and they continued to look a threat on the counter-attack; Wilson thought he had double the visitors' advantage in the 77th-minute, but his left-footed strike curled just inches wide.
Moments later, Man United missed arguably their best chance of the match when Maguire towered above the Fulham backline to mee Eriksen's corner, but he could only thump his header over the crossbar from just a few yards out.
However, Maguire made amends in the 89th minute and Old Trafford erupted when the defender slotted home a first-time finish from inside the six-yard box to restore parity after Leno could only parry Fernandes's shot straight to the Red Devils No.5.
It was suddenly backs against the wall for Fulham as Man United pushed for a winner in the nine minutes of added-on time, but out of nowhere they managed to burst free through substitute Adama Traore, who sprinted past a couple of players on the right before picking out Iwobi, who took a touch to set his shot which nestled into the bottom corner, sending the Fulham fans into pandemonium.
Fulham will look to build on their victory at Old Trafford when they play host to Brighton & Hove Albion next Saturday, while Man United shift their focus to an FA Cup fifth-round tie against Nottingham Forest at the City Ground on Wednesday.