Arsenal were beaten on penalties after the scores finished 1-1 following extra-time. United also had Diogo Dalot sent off midway through the second-half.
Amorim said, "Thankfully the players were there, they fought. Sometimes what we talk about before the match is not tactical or technical, it's this. We have to compete no matter what, no matter what the situation we competed today. We deserved to win, Arsenal didn't deserve to lose but we deserved to win.
"The way we played, the way we fought was really tough since the beginning. We didn't have the ball too much but we had control without the ball.
"I want to also say thank you to all the fans. It was unbelievable. I did not understand that all that area was our supporters, so it was a good day for us now we have to rest the players, I'm already thinking about the next game they're really tired.
"Sometimes we have the feeling that everything in the end is going to be okay. I felt that since the beginning of the second half, even after the penalty. I think the Arsenal team felt that maybe today is not their day. I'm really pleased for the way they controlled the game without the ball. We defended very well, set pieces we were really strong. It was a good day but now I'm thinking about the next game and worried about the next game.
"We were facing a great team and it was not just 90 minutes, it was more than that and then you feel like the players are really tired. They put in new players, they controlled the ball all the time, it was a sacrifice for my players. But we deserved to pass through this round because we suffered all together and we showed character and that's a very important thing."