Christian Pulisic opened the scoring after four minutes before Adrien Rabiot doubled Milan's advantage. Alessandro Bastoni pulled one back for Inter Milan, but Diego Moreira restored the two-goal cushion before half-time. AC Milan held on after the break to secure a significant victory at San Siro.
Amorim was delighted with the result and the atmosphere, but insisted one derby victory would not transform the team's season.
"To have incredible nights like this, and when I say incredible nights I mean both the result and the atmosphere, is fantastic. We will also have difficult nights like the one against Bologna," he said.
"Things do not change from one day to the next. There is no button you can press to turn everything around immediately. It was simply an excellent evening and three points gained, just as we lost three points against Bologna.
"We have to be extremely honest with everyone. When you win these kinds of matches, you still need to analyse the situation with complete clarity. There will be nights like this, we will have good moments, but there will also be games when we do not play well and do not win. It is a process."
The Milan coach also offered his support to the players, insisting that their commitment had never been in question.
"I am really proud of my players. I said that after the last match as well, when we lost. We are taking it one game at a time. We have had some players unavailable, and I know they are doing everything they can.
"They are giving everything, and that is the only thing I ask of them. They will always have me alongside them and they will have my protection as long as they continue to give everything. Sometimes we will win, sometimes we will lose, and sometimes we will play badly, but they are working hard."
Amorim also praised the atmosphere at San Siro, describing it as the best derby he had experienced.
"I did not expect an atmosphere like that. It is the best derby I have ever played. The environment is truly unique, and I am very happy to have won it."
Inter coach Chivu accepted that his side had not shown the necessary determination, particularly in the way they approached key situations.
"Compared with last year, there is something we need to address in our approach to matches, in the superficiality and lightness with which we deal with certain situations. We lack a little aggression, determination and awareness of danger," he said.
"We have to find those things again. We need to understand that, at this moment, despite having a few more points than last year, we have lost something that we absolutely must recover."
Chivu explained that injuries and illness among his forwards had influenced his tactical decisions, but refused to use those problems as an excuse.
"We were forced to find a solution and make the best of the situation by changing the system. We knew that, in this type of match, it would be right to have pace out wide and attack the defensive line. Unfortunately, sometimes things do not work."
The Inter coach also urged his squad not to allow one defeat to undermine the work of the season.
"We should not throw everything away. We have reached the tenth matchday and this is the first game we have lost. We have 25 points, more than we had at the same stage last year. We have always known how to react and keep our feet on the ground. We want to build another important season, and we have to find the way to do that again."