The Cottagers won 4-3 in the shootout after a 1-1 draw that saw them take the lead through Calvin Bassey's header.
United equalised in the 71st minute thanks to an excellent finish from captain Bruno Fernandes to take it to extra time.
Penalties were then required, with Fulham goalkeeper Bernd Leno first saving against Victor Lindelof from 12 yards.
Zirkzee was then tasked with keeping United in the tie but also saw his penalty saved, with Fulham heading into the quarter-finals.
The Dutchman looked emotional in the moments after and appeared to wipe tears from his eyes with his shirt.
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United, who beat rivals Manchester City in the final last season, will not be returning to Wembley next month.
Whilst exiting the FA Cup as the holders is another blow to United's season, a damning statistic makes for more worrying reading.
Ruben Amorim's side have now conceded 13 goals from corners in all competitions this season, with Bassey's opener for Fulham the latest.
This is the most out of any team in the Premier League.
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United therefore continue to show frailties in defence alongside a lack of goals going forward.
Defeat to Fulham has also seen them fail to win back-to-back titles in the FA Cup, with Arsenal the last to do so in 2015.
Having also exited the Carabao Cup in the quarter-finals, the Europa League is their only realistic chance of winning a trophy this season.
Victory would see them face either Lyon or FC Steaua Bucuresti, who were their final opponents in the league phase, in the quarter-finals.
One of Fenerbahce, Rangers, Roma and Athletic Bilbao will then be their semi-final opponents should they progress that far.
In the Premier League, United continue to languish in the bottom half of the table as they sit 14th.
There are 13 points separating them and the European qualification places with 11 matches left to play.