Matt O'Riley was forced off with an ankle injury just minutes into his debut after a late challenge, while Crawley also had Jack Roles sent off late on. Brighton won via goals from Simon Adingra, Jeremy Sarmiento, Adam Webster and Mark O'Mahony.
Hurzeler later said: "On the one side I'm not really happy about the performance and how the game went. I think it was too wild, too many mistakes with the ball, too shy, not enough courage against the ball, too passive, pressing individually. And the second part is you have to win tough games like this which we did.
"I experienced a lot of cup games in Germany and I know how difficult it is to play against a team that comes from a lower league. I think they had nothing to lose, they had a lot to win. So in the end, we are happy about the result but we will try to keep improving and try to play better than today.
"I think after this foul (on Matt), it was like the players thought they could tackle like they want because he didn't get a yellow card for this foul. For me it was a clear red card. You can't tackle like this, in my opinion.
"You can play hard. You can be ruthless. You can be hard in individual duels but you can’t the risk to injure a player and I think in this situation (the tackle) was much too late. I hope Mats is not so badly injured like it looks but we wait for the scan and then we can give an answer on that.
"In the end, we have to accept how the opponent play. And of course, it's not nice if a player is injured, but that was their way of playing and we have to find a way to deal with it, and that's something where we have to take responsibility on our own. We have to be honest. We have to improve in these things, and I'm sure we will."