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 Fulham won 4-3 in the shootout and advanced to the quarter-finals.
United had to come back from 1-0 down at halftime after Calvin Bassey headed Fulham in front just before the break, before Bruno Fernandes levelled the game to send it to extra-time.
Earlier, former United striker Danny Welbeck fired Brighton into the last-eight with an extra-time winner against Newcastle.
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 shootout, Welbeck produced a brilliant finish in the 114th to lift the ball over Newcastle goalkeeper Martin Dubravka from a narrow angle.
The 34-year-old former England forward had timed his run perfectly to beat Newcastle's 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.
Newcastle had led 1-0 through Alexander Isak's penalty in the 22nd, but Yankuba Minteh leveled 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.
Fabian Schar thought he'd scored a late extra-time winner for the hosts but it was ruled out for offside by VAR.
Welbeck's goal secured a fifth-straight win for Brighton in all competitions.
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, the lowest-ranked side left in the competition, will be at home to Premier League 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 on weekend of March 29-30)
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