Few gave Amorim's men much hope heading to face the runaway Premier League leaders at Anfield after a four-game losing streak.
However, the visitors were a side transformed from meek defeats to Bournemouth, Wolves and Newcastle as they scored their first goals away at Liverpool since 2018 and merited a share of the spoils.
"I want to see my team play so much better but the key today is the mentality," said Amorim.
"I feel that we are, not just the players but everyone at Manchester United, too comfortable. We need sometimes a shock and you can see today we were a different team."
United captain Bruno Fernandes said he was "upset" by the fact his side showed what they are capable of against the Premier League's best but have consistently underperformed this season.
"Today we were a different team not because of the system or tactical part of the game. It doesn't matter who the opponent is, we need to play every day like that," added Amorim.
"Today is all about the focus. The focus on set pieces, the focus on passing the ball where we cannot lose the ball because in the other part of the pitch is (Mohamed) Salah and (Luis) Diaz.
"If you have that mentality every day we are a different team."
Liverpool have a six-point lead at the top of the table with a game in hand over second-placed Arsenal, who were also held to a draw by Brighton on Saturday.
But Arne Slot conceded his side had missed an opportunity to streak further clear and close in on equalling United's record tally of 20 English top-flight titles.
Lisandro Martinez smashed United in front early in the second half.
But Liverpool rallied to lead through Cody Gakpo and Mohamed Salah's penalty before Amad Diallo levelled.
Harry Maguire should have secured United's first win at Anfield since 2016 when he skied over with virtually the last kick of the game in a frantic finale.
"Of course it feels for us like two points dropped," said Slot.
Both United goals originated down Liverpool's right as Trent Alexander-Arnold suffered a torrid time before being replaced late on by Conor Bradley.
Alexander-Arnold has been linked with a move to Real Madrid with the European champions reportedly having a bid for the England international turned down this week.
Slot, though, dismissed suggestions that the speculation was the source of Alexander-Arnold's trouble.
"It was clear Trent had some difficult moments, as we had as a team," added the Dutchman.
"What affected him is that he had to play against (Diogo) Dalot and Bruno (Fernandes) and that tells you how much quality United have.
"That is what United, once in a while, does and it is very difficult to play against them."
