Saints once again put their Premier League struggles to one side and shone in the cup, following up wins against Shrewsbury, holders Arsenal and Wolves, by stopping the Cherries from making history.
Redmond ruined Bournemouth's hopes of making their first ever FA Cup semi-final, unlocking the Championship side's defence with a superb ball for Moussa Djenepo's first-half opener at the empty Vitality Stadium.
The 27-year-old then produced a brilliant solo run and finish similar to Michael Owen's goal against Argentina at the 1998 World Cup in first-half stoppage time, before he completed a comprehensive 3-0 quarter-final win.
"I'm definitely happy, not only for coming to the next round but also for seeing a performance from my team that was very lively and by far one of the better ones in the (recent) past," Southampton boss Hasenhuttl said.
"I said in January that now is the time to take this chance because it's always a good chance for a smaller club like we are to write history.
"I think if you're good in the league and we had been in a very good position at that moment, then you can concentrate on the FA Cup.
"We took the risk, that cost us a few injuries in January, February. This was the problem then because we were always going with the strongest side against Wolves, Arsenal to go there and to go through.
"But finally now we get the reward for it and we have now a very, very good game in Wembley.
"It's a highlight for us and it will be, for sure, a strong opponent waiting there for us. But with one game everything is possible.
"I think the message I sent during the week was a very clear one, how important this game is for everybody.
"I think the players showed that there was no doubt that they are leaving everything on the pitch today."
While nobody put a foot wrong for Saints, Redmond was the undoubted star man as he displayed the kind of quality that has been too often missing from his game this campaign.
"He had a very difficult season so far, to be honest," Hasenhuttl said of the former Birmingham and Norwich winger.
"I think he's a very sensible player and he sometimes needs this one or two actions in the beginning, these positive ones, and then something changes in his mind and then he believes in what he's doing.
"This is what he should do more often, I think, and he knows he can do and can deliver this more often.
"But it's important for him also for the belief, what's he doing, that he only gets this game if he invests a lot.
"I think physically he's now in a better shape and can do more, and then finally you deserve more then."
Bournemouth boss Jonathan Woodgate was annoyed by Redmond's goal in first-half stoppage time, saying the throw-in that led to it should have been theirs and was a "blatant" wrong decision.
But the former England international knows the Championship play-off hopefuls were second best on Saturday lunchtime, when the video assistant referee ruled out an early Cameron Carter-Vickers own goal and second-half Che Adams strike.
"They're a Premier League team, we were never expected to win the game," said Woodgate, who was named head coach for the remainder of the season following Jason Tindall's exit last month.
"They were better on the day, more clinical in both boxes, really. They're a good team.
"Gutted that we've gone out. It would have been a bonus if we had got to a semi-final but the objective this season is to get into the play-offs and get promotion.
"We've got nine massive games to come so we'll put that game to bed and we move on."