Saints moved top of the Premier League for the first time in their history with a 2-0 win over Newcastle.
Che Adams gave Southampton an early lead and Stuart Armstrong wrapped up all three points in the 82nd minute to put Hasenhuttl's men top of the table ahead of this weekend's fixtures.
The south coast club last stood at the summit of English football 32 years ago under the management of Chris Nicholl and it marks a remarkable 12 months for the Austrian manager since their 9-0 thrashing by Leicester last October.
Cascarino is a firm fan of Argentine boss Bielsa, who led Leeds to the Premier League playing a specific brand of football which has not relied on lucrative transfers - instead choosing to improve the squad he has at his disposal.
And the former Chelsea striker insisted on the Weekend Sports Breakfast both Southampton's board and Hasenhuttl deserve a tremendous amount of respect.
He said: "What you did see, is a board actually backing their manager, knowing what is going on in the camp.
"They are privy to the camp, knowing what he is doing, the work he is doing, improving the team and I think they need to take huge credit.
"And the players, one thing that has never happened - even with all that disappointment of results - is they have never any element of discontent at Southampton.
"We didn't hear stories about players or player power going on - it just didn't feel that way.
"He seemed to keep it in tow, he managed the players and, with that, he made players better.
"I compliment Bielsa all the time on how he improves players, Ralph Hasenhuttl has done exactly that at Southampton."
Hasenhuttl played down the significance of their league position after they moved top of the Premier League for the first time in their history.
"Maybe it's surprising, also for us," said Hasenhuttl. "But no, when you look how much we've invested, the time and how good the game is at the moment - any part of the game - I don't think it's so surprising at the moment.
"We have definitely shown again today that we have every part of our game developing and this is I think the reason why the performances now (are as good as they are)."
Tony Cascarino believes Ralph Hasenhuttl deserves to be compared to Leeds United boss Marcelo Bielsa for the work he has done with Southampton.