Sunday's game survived a pre-match pitch inspection after heavy bouts of rainfall had left the field in a precarious condition, with Andre Marriner giving it the go-ahead an hour before kick-off.
Fulham enjoyed the better of the early possession, but it was the hosts who took a surprise lead on the half-hour mark.
A surging run from deep by Harry Maguire cut open the Cottagers' defence and he delivered a low ball across goal where Chris Porter was waiting to stab it past David Stockdale.
Almost immediately, there was a penalty appeal at the opposite end as George Long looked to have clipped Hugo Rodallega, but Marriner waved away the protests.
The hosts had Michael Doyle shown a straight red for an altercation with Chris David and Fulham were able to capitalise on their one-man advantage as Rodallega levelled with a fine strike from the edge of the area.
Rene Meulensteen made his priorities clear, making 11 changes for the game but he would still have hoped for better than a laboured 1-1 draw against a side who are not yet safe in Sky Bet League One.
After a late pitch inspection, Fulham's day began badly.
John Arne Riise's involvement lasted just 10 minutes, the full-back injured as he slid to dispossess Flynn at the byline.
With rain continuing to fall, there was little chance of flowing football, Pajtim Kasami and Rodallega coming closest before former Everton man Jose Baxter bent his free-kick wide.
Another heavy downpour threatened to make life even more difficult for the players but just after the half-hour, the Blades gave the game the goal it needed.
Maguire was the unlikely creator, advancing up the field before powering a low ball into the box where Porter was able to get the crucial touch and poke it home.
The celebrations would have been short-lived had Marriner awarded Fulham a penalty a matter of seconds after the restart but he was unmoved by Rodallega's appeal.
The striker was one-on-one with Long after connecting with Muamer Tankovic's through ball and appeared to be clipped on his way through.
As it was, the lead held until the interval and within 90 seconds of the second half, United were without both men involved in their goal.
Porter was replaced at half-time by new signing Stefan Scougall and Maguire followed him in making an early exit after collapsing in pain for the second time in the match.
The disruption was to get much worse for United in the 53rd minute when Doyle lashed out at David off the ball and was dismissed without hesitation. David's reaction was theatrical to say the least, but the lack of dissent from Doyle suggested he had erred.
Remarkably, the sending-off was almost followed by the hosts' second goal as Flynn broke at pace, shrugged off Hughes and drilled low at the near post. Fortunately for Fulham, Stockdale stood firm and saved with feet.
The addition of Darren Bent from the bench at least gave Fulham a proven scorer on the field, while the mercurial Adel Taarabt joined the fray with 24 minutes remaining.
But the League One side would not lie down and with 70 minutes gone, Flynn closed in on goal, only to be tripped by Hughes.
Marriner again waved away appeals, seemingly incorrectly. It was a massive call and the visitors cashed in with 15 minutes remaining.
It took a single moment of quality from Rodallega, who shuffled into space 20 yards from goal and fired low into the bottom corner.
Philippe Senderos then rattled the woodwork and the centre-half was involved again in a move that saw Bent fluff his finish after Kasami nodded across goal.
The Blades survived a late onslaught of pressure and a replay will now take place at Craven Cottage on February 8.