Hasenhüttl's side led 1-0 through Nathan Redmond's early goal at the time of Yan Valery's 29th-minute dismissal, before Swansea drew level 13 minutes from time and then took the lead early in extra time.
Saints rallied to equalise when Shane Long set up fellow substitute Mohamed Elyounoussi, before Long's first Saints goal in nearly two years secured a thrilling victory.
“In the end it was only hard work today, and something that we didn't want to have in the first round, for us, of the FA Cup," Hasenhüttl said.
“We have come to the next round, which is good, and we wanted to stay in this competition. You could see it – we were fighting for it and the team did a very intense, good job.
“I must say this season we have a very good atmosphere in the team. We are absolutely growing and I think we have a dressing room that is very, very committed.
“In such games it helps massively, because they don't want to lose – they want to fight and this is what you need in such a game."
Need a VPN? Or seeking to switch? Then support Tribal Football by clicking here to sign up with our official partners NordVPN. Not only will you receive the best VPN experience, but also great discounts by signing up via Tribal Football. Thanks for your support!