The Cottagers took a shock lead through Steve Sidwell but were unable to keep United out as they fell under ever-increasing pressure and were beaten twice in the space of as many minutes by Robin van Persie and Michael Carrick.
Rene Meulensteen's men refused to throw in the towel, though, and grabbed a stoppage-time leveller through Darren Bent to pile more pressure on beleaguered United boss David Moyes.
Fulham took the lead on 19 minutes when Sidwell was allowed to drift untracked down the centre of the field and cushion a perfectly-weighted pass from Lewis Holtby past David de Gea and into the bottom corner.
The visitors were then faced with a United onslaught, which saw them pepper the penalty area with crosses, and their resistance was finally broken 12 minutes from time when Van Persie converted from close range.
Carrick then steered a 20-yard drive into the top corner two minutes later to turn the game on its head, but there was to be late drama as De Gea parried a Kieran Richardson shot and Bent stooped to turn the ball into an empty net.
For Moyes and United, now nine points away from a Champions League place, a hellish season just keeps getting worse. And, with a visit to Arsenal on Wednesday, there is no end to the trouble in sight.
Moyes admitted in the build-up that Meulensteen knew more about the home players than he did, something that can only be expected after spending six years as Sir Alex Ferguson's assistant until last summer.
What was less obvious was the Dutchman's method of taking advantage of that inside knowledge.
In axing Scott Parker and Brede Hangeland and placing his faith in 18-year-old debutant Muamer Tankovic and midfielder Ryan Tunnicliffe, he was injecting his side with vitality so obviously lacking over an eight-game run that brought just a single win.
Holtby was the inspiration behind the visitors' opener. Allowed far too much room to advance down the left flank, the German playmaker was able to drop a superb cross into the heart of United's penalty area.
Nemanja Vidic had been pulled out of position and as neither Darren Fletcher nor Juan Mata had tracked Sidwell's run, the midfielder was able to slide the ball past a stranded De Gea with a degree of comfort.
It was the first time Fulham had scored an opening goal since December - and they should have scored more.
Holtby created a golden chance for Richardson but the former United man blazed over.
Then Richardson scorched past Vidic, only for Wayne Rooney to intercept a cut-back that looked certain to set up Holtby.
United were creating chances. Maarten Stekelenburg superbly denied Carrick's thunderbolt and a powerful Vidic header. Van Persie failed to snaffle a half-chance at the near post.
But that attacking instinct that has terrorised so many opponents down the years was missing from the hosts' game.
Adnan Januzaj's introduction for Fletcher, followed by the arrivals of Javier Hernandez and Antonio Valencia, meant United had six offensive players trying to score against a team that had kept just four clean sheets all season.
Really, United could not fail to score.
It was noticeable the accompaniment to Van Persie tucking home Juan Mata's cross at the far post was merely to lead a charge back to halfway, knowing the job was only half done.
When Carrick's shot looped in off Parker a couple of minutes later, it seemed something was finally going to go Moyes' way.
The Scot should have known better.
Bent will rarely get an easier goal when it came in the fourth of five added minutes, and while the full-time boos were half-hearted, there really is little positive that can be said of Moyes' reign right now.