The victory, sealed by a late Harry Maguire header assisted by Bruno Fernandes, ended United's nine-year wait for a league triumph at Anfield and handed Liverpool their fourth successive defeat in all competitions.
United struck inside the second minute when Bryan Mbeumo finished from close range after a lively run from Amad Diallo.
But Liverpool dominated large spells, hitting the post twice through Cody Gakpo and squandering several clear chances, including a one-on-one opportunity for Alexander Isak that was denied by debutant goalkeeper Senne Lammens.
"They had two shots on the post that could change the narrative," Amorim said. "Sometimes we had a bit of luck, but the goalkeeper showed a lot of character. It was not a very well played game from our side, but the spirit was great."
Gakpo eventually levelled for Liverpool in the 78th minute after a pass from Federico Chiesa, but Maguire rose highest to head home the winner six minutes later, a fitting redemption after missing a late chance in the same fixture last season.
"Harry Maguire was really important for us today," Amorim added. "He's a great example after everything he has been through."
Liverpool manager Arne Slot admitted his side's wastefulness in front of goal was a key factor in the defeat, with the result leaving them fourth in the table amid a run of four straight losses.
United climbed to ninth but Amorim played down talk of a Champions League push, insisting: "We just need to win three games in a row. Forget about the top four.
"The way people see the team is different, but we are the same team."
Harry Kane, Lamine Yamal and Kylian Mbappe headline the 30-player shortlist for the 2026 Ballon d'Or, with the winner to be announced at the ceremony in London on October 26.
Manchester United have reportedly decided against bringing David Alaba to Old Trafford, despite the Austrian defender being available as a free agent.
Liverpool midfielder Alexis Mac Allister has admitted he was "very, very sad" after being told the club was not in a position to offer him a new contract.
Manchester United supporters have been urged to come together for a peaceful demonstration before Sunday's derby against Manchester City at Old Trafford.
