Victor Lindelof and Joshua Zirkzee had spot kicks saved after the fifth-round game had ended 1-1 after extra time at United's Old Trafford home on Sunday.
Fulham's German goalkeeper Bernd Leno was the hero as they advanced to the quarter-finals.
"I know they had chances but I think we were the best team on the pitch," Cottagers coach Marco Silva said.
"Our team was on the ball, dominating the ball. We deserve to be in the next stage."
United had to come back from 1-0 down at halftime after Calvin Bassey headed the visitors in front just before the break, before Bruno Fernandes levelled to send the game to extra time.
"The goal is to win the Premier League. So I know that we are losing games and losing trophies during this season, but the goal is to win the Premier League again," said United coach Ruben Amorim.
"I don't know how long it's going to take. But that is our goal.
"If you think like that, we can suffer a little bit in these defeats — a lot sometimes — but we have a goal, so we continue, no matter what, we continue forward."
Earlier, former United striker Danny Welbeck fired Brighton & Hove Albion into the last eight with an extra-time winner against Newcastle United.
Welbeck, a two-time winner of the famous trophy with Arsenal, came off the bench to seal a 2-1 win at St. James' Park.
With the game locked at 1-1 and seemingly heading to a penalty shoot-out, Welbeck produced a brilliant finish over Newcastle goalkeeper Martin Dubravka in the 114th minute.
The 34-year-old former England forward had timed his run perfectly to beat the offside trap and meet Solly March's through-ball. Despite running away from goal, he managed to turn and clip his shot into the back of the net.
The Magpies had led 1-0 through Alexander Isak's penalty in the 22nd minute, but Yankuba Minteh levelled in the 44th.
Both teams had players sent off in the second half — Anthony Gordon for Newcastle in the 83rd and Tariq Lamptey in added time.
Newcastle are hoping to end a 70-year wait for a major domestic trophy and can still achieve that this season when they play Liverpool in the League Cup final later this month.
In the draw for the quarter-finals made later, Fulham will host London rivals Crystal Palace, second-tier Preston North End, the lowest-ranked side left in the competition, will be at home to Aston Villa.
There's a tasty all-Premier League clash as Manchester City head to Bournemouth, while Brighton's reward for the Newcastle win is a home tie against either Nottingham Forest or Ipswich, who play on Monday night.
FA CUP QUARTER-FINAL DRAW:
Fulham v Crystal Palace
Preston North End v Aston Villa
Bournemouth v Manchester City
Brighton and Hove Albion v Nottingham Forest or Ipswich Town
(Ties to be played weekend of March 29-30)
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