The hosts took the lead early on through a fine finish from Ademola Lookman, but Edinson Cavani pounced to level the scores and Paul Pogba won the game for the visitors in the second half with a superb strike from outside the area.
Story of the game
Fulham boss Scott Parker made two changes to the team that suffered a narrow to defeat to Chelsea at the weekend, both of them enforced: Joe Bryan and Ruben Loftus-Cheek started in place of Antonee Robinson and Bobby Decordova-Reid who were unavailable due to suspension.
Meanwhile, Ole Gunnar Solskjaer made three changes to the starting 11 that held champions Liverpool to a draw last time out, with Cavani, Mason Greenwood and Eric Bailly replacing Marcus Rashford, Scott McTominay and Victor Lindelof.
Their commitment to high pressure made it a bright start for Fulham; within two minutes, Ivan Cavaleiro dispossessed Pogba in the middle of the park and fed Lookman, who drove forward to the edge of the area but eventually gave David de Gea a comfortable save.
That bright start soon became a perfect one. Andre-Frank Anguissa picked his head up in midfield and spotted the run of Lookman, who found himself in acres of space when he received the ball, with just de Gea to beat; the forward made no mistake whatsoever with a deft finish.
United appealed for VAR to intervene, but Lookman had been played well onside by Aaron Wan-Bissaka. This was going to be a test of character for the team hoping to end the evening top of the table.
The hosts didn't sit back intentionally, but their visitors had them pinned near their own penalty area for a period of about 15 consecutive minutes, probing for that opening to level the scores. To no avail, though; Fulham's lead looked to be remaining comfortably intact.
United had enjoyed not a single good opportunity in the opening 20 minutes, and then two came along at once. First, Bruno Fernandes rattled the upright with a vicious strike from the edge of the box. Then play was recycled for the 26-year old to cross, and Alphonse Areola palmed the ball straight to the feet of Cavani who could make no mistake from about eight yards out.
Suddenly the visitors had flicked a switch, and it all seemed to be coming through their star Portuguese midfielder. Fernandes tried his luck from 20 yards again, and this time drew a super save from Areola at full stretch.
The attacking intent of United didn't relent. Looking to atone for his earlier error, Wan-Bissaka darted towards the byline before crossing towards the back post where Anthony Martial rose above Kenny Tete but failed to direct his header goalwards.
Fulham conjured an excellent chance to regain their lead on 37 minutes when Anguissa, the architect of the first goal, thread Cavaleiro through on goal, but the forward's poor first touch ran the ball straight out of play.
After a period of frailty, the Cottagers had rediscovered their footing in proceedings, while United had done well to even up the scoreline but the lead continued to evade them; the final chance before half-time saw Martial head over from another Fernandes cross from the left, and the match went into half-time deadlocked.
Fulham's energy seemed to have waned towards the end of the first-half, but they took to the restart as they had to the beginning of the match, pressing high and fashioning the opportunity for Harrison Reed to strike from the edge of the box, but the effort was no where near troubling de Gea.
They started having a lot of joy down their right, with Loftus-Cheek heavily involved. He twice skipped past Luke Shaw, first supplying an inviting driven cross which was just too far in front of Lookman, then cutting the ball back for the goalscorer in the next phase; this time the forward's effort was saved comfortably by the goalkeeper.
For the most part, though, the deadlock continued. Both sides had their spells, but this was an even contest - United enjoying the majority of the possession, Fulham conjuring some decent opportunities on the break.
Until the 64th minute, that is, when a moment of brilliance gave the visitors the lead. Pogba skipped past Reed's challenge and took aim on his left foot - it was a flat shot with tremendous power and a bit of curve, and it left Areola completely helpless.
Fulham's French shot-stopper was equal to his next challenge, though; an inch-perfect cross from Fernandes picked out Cavani centre of goal, and the Uruguayan's close-range header drew a fantastic reactionary save from Areola at his near-post.
The hosts were lucky not to see the game slip away from them in the space of a couple of minutes and they were still in this game - just.
And Fulham could have levelled the scores with 75 minutes on the clock when substitute Aboubakar Kamara played Loftus-Cheek behind the United back line, but de Gea did well to save the low shot with his legs.
Continuing to search for their equaliser, Reed floated in a superb free-kick for Fulham which caught both de Gea and the United defence in no man's land, yet no one was there to latch onto it.
Moments later, they fashioned their best opportunity of the half. Loftus-Cheek bolted forward and passed wide for Lookman who in turn crossed into the centre where Aleksandar Mitrovic was lurking. The Serbian skewed his header but it fell straight back to Loftus-Cheek; he blazed over from 15 yards.
The closing stages belonged to the hosts, and there was one more edge-of-your-seat moment before the match ended. Tete swung in a cross which found the head of Mitrovic and the striker's effort deflected off the thigh of Bailly and looked agonisingly close to trickling in with de Gea stranded, but it bounced just wide of the near post.
United held strong for the final few moments and, just like that, they had secured a vital three points to send them to the summit of the standings.
Takeaways from the match
Energy is a finite resource, but Fulham defy it
Parker was aggrieved when his match against Tottenham last week was rescheduled at short notice, giving him limited time to prepare. It meant he would need to commit to fielding a full-strength side for a long run of consecutive matches - they can't afford to give anything but 100% in their league exploits with relegation still a real possibility.
You'd have forgiven Fulham for being tired, then, but there was no evidence of that here. They pressed relentlessly and countered with pace and enthusiasm, and some may argue they were good enough value for a draw here, but quality prevailed in the end.
Digging deep sends United top
Falling behind early was a shock, and it provided United with a challenge which they didn't look prepared for in the opening stages. Throughout the night perhaps it didn't click exactly how they would've liked, but the new league-leaders managed to earn victory the hard way.
Cavani's equaliser came after a good 20 minutes of patient probing, while Pogba's winner was a moment of individual brilliance to seal the result. United rode their luck to some extent in the last few moments - all they'll care about, though, is those precious three points.
Fulham: Areola; Aina (Mitrovic 83'), Andersen, Adarabioyo; Tete, Reed, Anguissa (Lemina 79'), Bryan; Loftus-Cheek, Cavaleiro (Kamara 71'), Lookman.
Subs not used: Rodak, Odoi, Hector, Ream, Onomah, Kebano.
Manchester United: de Gea; Wan-Bissaka, Bailly, Maguire, Shaw; Pogba, Fred; Greenwood (McTominary 85'), Fernandes (Matic 90'), Martial (Rashford 85'); Cavani.
Subs not used: Henderson, Tuanzebe, Telles, Matic, van de Beek, Mata, James.
Following defeat here, Fulham are at home again at the weekend as they face Burnley in the FA Cup.
United, meanwhile, return to Old Trafford to host Liverpool in the same competition.