Leeds won on penalties after Axel Disasi had brought the scores level for West Ham deep into injury-time.
Nuno remarked afterwards: "It was a very emotional game from the beginning till the end.
"For the boys, it's sad, it's sad, it's sad for us, especially for our fans, the way they saw the second half, us reacting, chasing the game, creating the best chances, hitting balls in the post.
"I think we did enough to avoid the penalty shootout, but today was not to be.
"In the previous round we came through to the penalty shootout and we said that we were efficient, but today we were not so efficient.
"But the way we reacted in the game itself, the way we chased the game, it's the proof that we don't give up and we have a big, big mission on our hands (in the Premier League)."
He also said: "Now it's time to recover. It's 120 minutes, it's tough. We'll try to recover our players and play the game on Friday.
"Again, London Stadium is going to be so important for us. The atmosphere that we created today, we have to create it for the game, for the matches moving forward.
"We need it, we need it, we need it. We need all the players. That's the proof."