The Red Devils scraped past their Premier League rivals to progress into the fifth round of the competition, as they prevailed 2-1 at Old Trafford thanks to Harry Maguire's last-gasp header.
Despite victory however, it was another lacklustre performance from United, who trailed at half-time after Bobby Decordova Reid had put the Foxes ahead.
Joshua Zirkzee equalised on 69 minutes, before Maguire headed home a stoppage-time winner, despite appearing to have been offside in the absence of VAR in the fourth round.
And head coach Amorim was clearly less than impressed by his side's narrow victory, who took immediate action after full-time.
A post on United's social media channels showed members of their first-team taking part in a training session on the pitch following the conclusion of the evening match.
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Confirming the players were training, rather than doing a customary warm-down, the club wrote: "Working for the badge.
"Some of the United squad completed a full training session at Old Trafford following the full-time whistle on Friday."
And while many fans online were impressed by Amorim's hard-line approach, Arsenal legend Keown has slammed his methods.
Appearing on talkSPORT's White and Jordan, the Gunners Invincible accused the Portuguese coach of being unprofessional, and argued the training session had been a move to humiliate his players.
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Keown said: "Does it come across as a punishment though for the players?
"I mean he can do what he wants. It does feel a little bit unprofessional, [like]: 'Because we haven't got time in the week to do it, so we are doing a session afterwards'."
He continued: "It's very unorthodox. It's not the norm, there's recovery, there's stretching, okay? They've got nine days before they play their next game.
"But is it about embarrassing the players a little bit? Look, they were awful for large periods of that game.
"So for me it has to be [that] he's so annoyed, there's no time to wait, and no time to lose, because he could be out of a job as well, it's not beyond the realms of possibility."
Keown then suggested Amorim would have been better off addressing his side's problems in their nine-day break before their next match, a league clash with Tottenham on Sunday.
The 58-year-old continued: "But what's more professional is to make sure you find a better way to get the message across to your players in the time available, not necessarily immediately after a game.
"You've got some players who played a full 90 minutes taking part in that training session. I don't think that's ideal.
"So he's got to do better in the time that he has got available. He's got nine days between now and the next match, he could have waited but he didn't want to.
"That's how annoyed he was right? But they're not picking it up, and they're not enjoying this system, they don't look suited to it at all, and thank God for Harry Maguire, to be honest!"
United are currently languishing in 13th in the Premier League table, with Amorim winning just four out of 13 league games since his introduction as head coach in November.
And Keown admitted he struggles to see an immediate end to the 40-year-old's woes at Old Trafford.
He said: "I'm looking at results, I'm looking at performances and they're not getting any better.
"And it looks like the squad is shrinking, and some of the players that are on the bench should be starting.
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"But there's an argument they might not perform from the beginning, but that's not the answer is it?
"I'm just telling you the facts, as I see Man United right now, it's a club that's on the slide."