The visitors took the lead through Viktor Gyökeres before Stuart Armstrong's wonder strike drew Saints level shortly after the hour mark.
With extra time drawing to a close and the prospect of penalties looming, Kyle Walker-Peters netted to give Saints the advantage at St Mary's.
“This is what the FA Cup is about," admitted Hasenhüttl. “There's no easy games and it's definitely always a tough fight and we were against a very good, organised opponent today.
“We were coming after a two week break so you could feel in the first half we were lacking a little of automatism and quality.
“With the subs we became better and better and then we had a few good chances. It's not easy to prepare chances against them but in the end we struck two wonderful goals.
“It was a good fight and in the end there was a good goalkeeper save from Will so it was a tough fight today.
“Magnificent goal [from Stu]. He has a fantastic finish from left and right so when he's around the box he's a real threat. It was beautiful.
“[Kyle Walker-Peters] scores goals now and this is what I need from this position to be honest. He's always in the box and being dangerous there, it's very important for me and he showed he can do it also from the left side.
“The opponents have also been very loud today so it was a really, really intense atmosphere but that's how FA Cup games should be."
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