Chelsea twice came from behind before Alejandro Garnacho and Joao Pedro struck in extra-time to earn the 4-2 win in Wales.
"That was every reason why the FA Cup is what it is," said Rosenior.
"Wrexham were magnificent in their energy and how brave they were with their press and how they played. We had to be at a high level. We were pushed all the way by a very good team. We need to find a way to manage games better, but in these Cup ties you can feel the emotion of the game. They’re very difficult games to play in.
"We went 1-0 down early against Aston Villa, we were 2-0 down at half-time against West Ham, we have to find a way not to be in that position, but what the players are showing me, no matter who is on the pitch, is a really good spirit and belief and a determination to win games of football.
"Now we need to keep working, we need to recover because we’ve got a big game on Wednesday and we need to start preparing for that."
On Garnacho's performance in extra-time, Rosenior added: "Ale is in a great place.
"His goal was top, his energy levels for 120 minutes were magnificent considering he played a full 90 on Wednesday.
"He’s in a really good place and it’s really good to see him getting his form at a really important part of the season."