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 at Old Trafford.
Bruno Fernandes quickly put United ahead with his 100th goal for the club in all competitions in his 200th Premier League appearance.
Casemiro doubled United's lead before the Brazil midfielder was dismissed for a second booking in first-half stoppage time.
Trevoh Chalobah's late reply for Chelsea set up a tense finale, but United held on for just their second win in six matches in all competitions this term.
Having kicked off languishing just above the relegation zone, Amorim's side climbed provisionally to ninth place, to give the Portuguese coach a little breathing space.
"It was really important. I think we deserved it, but we complicated our lives every time," Amorim said.
"I think we started the game really well in terms of the aggression and to push the opponent and to show that we were there, even to bring our fans with us.
"Of course, the sending-off of the goalkeeper helped us a lot, but we were in control, we scored two goals. And with everything going well, we made things complicated with that sending-off.
"The second half, I think we managed well, but, in the end, the last 15 minutes was tough for us. They scored a goal, and then we suffered together and we managed to win."
After such a torrid start to the season, Amorim was proud of the way United dug deep to secure just their second win in all competitions this term.
United failed to win successive Premier League games last season, and Amorim challenged his players to end that barren run at Brentford next weekend.
Amorim demands urgency
"Sometimes we have some moments that we feel an urgency to have a result. This team has shown many times that we have that urgency," he said.
"Today we won. It's nice to win, but let's forget that and return to that urgency. We need to win the next game. That is the most important thing.
"Because in the big club it's not that feeling: 'today is a really good game, let's relax a little bit'. So, let's return that urgency."
Chelsea boss Enzo Maresca was left frustrated after his game plan was ruined by Sanchez's red card.
Asked if he would have preferred his keeper to have let Bryan Mbeumo through to score rather than foul him, the Italian said: "Yeah, probably it is the best solution because we are still 95 minutes to play.
"I think even Robert is aware of that. But also it's difficult because he has to take a decision in one second, two seconds. So, it's difficult.
"But, for sure, if you ask me, I prefer to be 1-0 down after three minutes than one player down."
Maresca made three first-half substitutions in a bid to stem the tide, including taking off England forward Cole Palmer, who had been struggling with injury.
"He had a test this morning to play this game. He was not 100 per cent fit," Maresca said after Chelsea's first league defeat this season.
"He made a fantastic effort to play this game, but he was not 100 per cent."