It came after this morning's dramatic fifth-round tie at Old Trafford ended 1-1 after extra time.
The match went to penalties after United's Bruno Fernandes grabbed a second-half leveller following Calvin Bassey's opener for Fulham in stoppage time at the end of the first half.
After a goalless extra time, Leno saved Victor Lindelof's penalty, leaving United's Joshua Zirkzee needing to bury his to keep their hopes alive. Yet he fired his effort straight at Leno sparking wild Fulham celebrations.
Raul Jimenez, Sander Berge, Willian and Antonee Robinson converted their penalties for Fulham, who will face Crystal Palace in the quarterfinals.
Fulham caught United's defence flat-footed from Andreas Pereira's corner to strike first blood seconds before halftime, with Rodrigo Muniz nodding on for Bassey to head home in his first FA Cup appearance.
Fernandes levelled in the 71st minute when Diogo Dalot cut it back from the left-hand side and United's skipper unleashed a left-footed shot from just inside the 18-yard box that sailed past two Fulham defenders and goalkeeper Leno on its way into the far corner to send the game into extra time.
"Unbelievable," a smiling Leno told the BBC. "I think we deserved to win after 90 minutes. Of course, penalties, it is sometimes a lottery and luckily we won. The goalie coach told me information, but my feeling is more important.
"I went the wrong way three times, but credit to our penalty takers - it is not easy. Brilliant job from all of us."
Thirteen-times FA Cup winners United defeated Manchester City last year and lost to City in the 2023 final.
But Ruben Amorim's men have struggled mightily this season, languishing 14th in the Premier League, five places below Fulham.
"I think we have the best chances in the game," Amorim said. "In the end, the penalties can go both ways and today wasn't our way.
"The goal is to win the Premier League," he added. I know that we are losing games but the goal is to win the Premier League again. I don't know how long it will take. We have a goal and we continue forward no matter what."
Both sides squandered chances at winning goals in the breathless affair at Old Trafford. United keeper Andre Onana dived to push away a shot from Emile Smith Rowe late in normal time.
United then had several terrific chances to seal the victory seconds before the whistle sounded. Leno came out to save a point-blank shot from Alejandro Garnacho and then 17-year-old Chido Obi first stabbed the ball just wide of the net and then forced Leno to make a diving save.
Fulham boss Marco Silva said his team deserved to win in normal time.
"I think we were the best team on the pitch. Our team was on the ball dominating the ball," he said. "We deserve to be in the next stage."
Fulham reached their only FA Cup final in 1975 when they were beaten 2-0 by West Ham United. Leno said he would like to see his team get back to Wembley for the final.
"Two years ago we lost here in the quarterfinal (to Manchester United) and we're in the quarterfinal again," he said. "We want to go to Wembley 100%."
Welbeck snatches win for Brighton
Danny Welbeck scored deep into extra time to send Brighton & Hove Albion into the quarterfinals with a dramatic 2-1 victory over Newcastle United, after both teams were reduced to 10 men.
Newcastle's Alexander Isak and Brighton's Yankubah Minteh scored in the first half, before Welbeck struck the winner in the 114th minute with an exquisite finish from a tight angle.
"It was a very emotional game," Brighton manager Fabian Hurzeler told the BBC. "Lots of ups and downs. The key was we stayed together, we stayed calm, we tried to find solutions."
Isak opened the scoring in the 22nd minute when he planted his penalty, awarded after Minteh fouled Tino Livramento, beyond the reach of goalkeeper Bart Verbruggen into the far corner.
It was Isak's 22nd goal in all competitions this season.
Minteh atoned for his mistake when he sprinted on to a threaded pass from Joao Pedro to equalise in the 44th minute.
Newcastle's Anthony Gordon and Brighton's Tariq Lamptey were sent off in the closing stages of normal time.
Gordon was shown a red card for his shove to the head of Jan Paul van Hecke while Brighton were reduced to 10 players in the 91st minute when Lamptey earned a second yellow card for a lunging tackle on Jacob Murphy.
Newcastle, six-times FA Cup winners, had two goals chalked off for offside in regulation time. Isak thought he had doubled the lead in the 32nd minute and Fabian Schar celebrated what he thought was the winning goal deep into stoppage-time before that was also ruled out.
"It's great to get the winning goal," Welbeck told ITV. "It was a solid team performance and there was a lot going on in that game today. Lots of ups and downs, but we stuck with it. I'm so proud of the lads.
"Newcastle can always put you under pressure. They are very dangerous, but we stuck to our guns and we put in a great shift. I am feeling very good, feeling fit and feeling strong."
Brighton's reward was a home quarterfinal against either Nottingham Forest or Ipswich Town, who meet on Monday.
Newcastle will now have to pick themselves up for the League Cup final against Liverpool on March 16 when they will attempt to win their first major trophy since 1955.
"It's not a great day for us, but it could have been -- we talk about fine margins in football, but if Fab's disallowed goal had been onside by the shortest of distances, we would be sitting here saying it was an amazing win and what great character the players showed," manager Eddie Howe said.
"I don't know if the cup final has been a negative influence on our performances of late, but I know the level has dropped from where it was a few weeks ago, and the most important thing is that we need to get them back up."
