Goals from Nick Tsaroulla, Ashley Nadesan and Jordan Tunnicliffe saw the Reds comfortably beat Marcelo Bielsa's side this afternoon.
Crawley manager John Yems, speaking to BBC Sport, said: “You have to enjoy these games – you work hard enough for it. It was a really good team performance and it's clear that we've got 11 star men.
"These players have got a lot to prove to the clubs who have released them and we’ve showed what we can do against a really good side in Leeds.
"Let’s see who we get in the next round and enjoy the moment."
Man-of-the-match Nadesan said the whole club 'has done everyone proud today'.
He added: "The first goal was a bit of a shock but to get the second goal straight after was really big. Personally, it's a great accolade to get man-of-the-match.
"We’ll take anyone from the Premier League here. We’re trying to make this place a fortress."
Tunnicliffe already knows what team he would like to come up against in the next round.
He said: "I'm a Baggies [West Brom] fan so they got knocked out, so maybe Manchester United at home.
"We lost in the Carabao Cup last season, and the team that beat us got United so we were gutted and we would love that chance."