Kiernan Dewsbury-Hall's superb first-half goal sealed a 1-0 win for the visitors.
Fighting to avoid the sack after a woeful start to the season, Amorim received a welcome boost when Chelsea goalkeeper Robert Sanchez was sent off after just five minutes of Saturday's clash.
A return to Europe's elite tournament is estimated to be worth at least £70 million ($93 million) to United and would allow Amorim more resources to rebuild his squad.
The Gunners sit 15 points behind runaway leaders Liverpool, now with just one game in hand, with their focus beginning to turn to holding off the chasing pack for a place in next season's Champions League.
United's third-placed finish in the Europa League first phase ensured they avoided a two-legged playoff to reach the last 16.
United crumbled to a sixth loss in 11 Premier League games since Amorim arrived from Sporting Lisbon to replace the sacked Erik ten Hag.
Ten Hag's position had been repeatedly called into question in recent weeks following several lacklustre results, with United also confirming Monday that Ruud van Nistelrooy will take interim charge.
Manchester City still have the destiny of the title in their own hands as the defending champions have two matches left to Arsenal's one.
Champions Manchester City are ominously poised to strike in the new year despite a dip in form from last season's treble winners, while Arsenal, Aston Villa and Tottenham are also in the mix.
Ten Hag's position had been repeatedly called into question in recent weeks following several lacklustre results, with United also confirming Monday that Ruud van Nistelrooy will take interim charge.
Manchester City still have the destiny of the title in their own hands as the defending champions have two matches left to Arsenal's one.
Champions Manchester City are ominously poised to strike in the new year despite a dip in form from last season's treble winners, while Arsenal, Aston Villa and Tottenham are also in the mix.
Elsewhere, Scott McTominay scored twice as Manchester United beat Chelsea 2-1 and Fulham hammered Nottingham Forest 5-0, heaping the pressure on manager Steve Cooper
Barca's friendly against Juventus, due to be played later on Saturday, was cancelled after an outbreak of viral gastroenteritis among the Spanish team
City captain Gundogan entered the record books as he scored the fastest goal in FA Cup final history, crashing home an exquisite volley after 12 second
United boss Ten Hag, speaking after the nail-biting Wembley shootout, praised his team's ability to bounce back
It will be United's first FA Cup final appearance since losing to Chelsea in the 2018 showpiece
Manchester United have signed Dutch striker Wout Weghorst on loan until the end of the season from Championship club Burnley, the Premier League side said on Friday.
Bruno Fernandes sent Manchester United into the League Cup last 16 as the Portugal midfielder punished a howler from Aston Villa keeper Robin Olsen in Thursday's 4-2 win.
