The Cottagers won on penalties at Manchester United on Sunday.
Silva said, "In some moments we showed very good quality, our identity was there. We sorted some things and did much better. we built our attacking moments really well.
"The way the team played in the second half and all of extra time makes me really proud because we came here and were the dominant team. We were brave, we did it in our way. It was a big fight. I love the composure of our own players.
On the penalties, he said: "The other guys, it's really important. They did it perfectly, with great composure from all of them. They are with a smile on their faces. Fulham fans are really proud of their team. We are there.
Asked if Fulham deserved to win in 90 minutes, the manager stated: "I think yes. I know they had chances but I think we were the best team on the pitch. Our team was on the ball dominating the ball. We deserve to be in the next stage.
On drawing Crystal Palace in the final eight, Silva added: "It is going to be a great battle between two good sides. They are in great form away from home. London derby."