The Argentine said last May he could not see a future for himself at St Mary's if his chairman were to leave the club.
But less than 24 hours after differences with Southampton owner Katarina Liebherr led Cortese to quit, Pochettino made it clear he would not be following.
He told Saints Player: "I am staying as manager. I have a professional responsibility to everyone at the club and the fans."
Pochettino then faced the press and was asked why his comments back in May no longer applied.
He said: "They are completely different situations. Eight months ago I was finishing my first season, and I did say it would make no sense for me to be here without Nicola.
"Now we're in the middle of our project and it's a completely different situation. I'm completely committed."
Pochettino confirmed he had spoken to Cortese about his decision but would not say whether it took a pep-talk from the Italian to persuade him to stay.
"What's quite clear is that I have a good relationship with him," he said. "He agrees with my decision to stay and there's nothing else I can say at the moment.
"He only asked me for one thing and that's to try and win on Saturday."
Cortese's departure has thrown the future of stars like Luke Shaw into doubt.
But Pochettino vowed no player will leave the club during the January transfer window without his authority.
"I want to make one thing clear: no one is for sale at Southampton, no one," he said.
"And I believe the same thing. No player that I don't want to leave will leave this club."
Pochettino said he wants to honour the remaining 17 months of his Southampton contract admitted it was too soon to say for sure at this point whether that will happen.
"It's very hard to say at the moment, I think it's very soon to say about that," said Pochettino.
"But when I signed with Nicola, and we signed a number of players, I'm 100 per cent committed to the two years of my contract and I'm 100 per cent committed to those players.
"I have a contract and a responsibility to that."
Some reports this week suggested Liebherr's desire to sell Southampton had contributed to a deteriorating relationship with Cortese, but Pochettino stonewalled when asked why his boardroom ally had resigned.
"I do not know the answer," he said. "It's difficult to say. I found out about his decision just as you did."
Questions about Liebherr's plans for the club and reports of a bid for left-back Luke Shaw met with a similar response, but the manager did reveal he retains the full support of the owner.
He said: "I spoke to her five minutes ago, and she transmitted her full trust, not just to me but to my staff and the people working here. We haven't had much time to think about the future."
And when pressed about interest from elsewhere in his players, he said: "I want to make one thing clear - no one is for sale at Southampton. No one. My viewpoint hasn't changed from a week ago. No player that I don't want to leave will leave."