They thought they were well on their way after Alan Smith put them ahead in the first half.
But then, with only three minutes to go, Diop unleashed an unbelievable and ferocious 25-yard shot that fizzed into the net to bring the scores level.
It was a great result for Fulham because they knew it was going to be a difficult night with first-choice players such as Luis Boa Morte, Lee Clark, Claus Jensen, Alain Goma, Mark Crossley and Moritz Volz all unavailable.
Manager Chris Coleman handed 20-year-old Liam Rosenior, son of Fulham legend Leroy, a surprise debut at full-back - and he was outstanding.
United, however, took control of the first half and the only surprise was that it took them so long to break the deadlock.
Roy Keane was the first to go close to breaking the deadlock, blasting a 25-yard effort full against Edwin van der Sar's left-hand post in the 29th minute.
Wayne Rooney then rattled the other upright, stepping neatly onto a Paul Scholes pass wide on the left and cutting inside to strike a shot against the woodwork.
Fulham, who were over-run in midfield and showing little up front, should have heeded the warning. But in fact they remained asleep when Cristiano Ronaldo fired a low ball across the area from the right wing after 32 minutes.
Sylvain Legwinski clearly thought he had time to clear the ball, but Smith slid in to win a big tackle - and got up to classily pass his shot into the net.
The goal was no doubt greeted with dismay at both Highbury and Stamford Bridge, and it was nearly 2-0 five minutes later when Keane saw another belter well saved by van der Sar.
In fact, the only thing that marred a good opening 45 minutes for United was a disgraceful dive from Ronaldo as he tried to win an undeserved penalty.
Fortunately referee Phil Dowd was not fooled - although inexplicably he decided no to book the Portuguese winger.
Ronaldo almost made up for his stupidity on the stroke of half-time, but dragged a good chance narrowly wide.
It meant Fulham at least went into the break only one goal down, and gave Coleman a chance to fire up his troops.
Whatever he said it seemed to work because within a minute they almost equalised when Diop's header from a corner was spectacularly tipped over the bar by Roy Carroll.
Then, with 51 minutes on the clock, Andy Cole twisted and turned to send a superb right-footed shot against the same post that had earlier denied Keane.
Those efforts at least gave Fulham and the home fans hope, although increasingly United regained control as the match wore on.
Fulham brought on Tomasz Radzinski and Collins John to try and grab an equaliser, and John immediately tried to claim a penalty for being held in the area.
Then Cole sent a back-header over the bar after Brian McBride had done well to win a high ball on the edge of the box.
It all seemed set for a 1-0 defeat. But then, with only three minutes to go, Diop produced his party piece to leave United stunned.