The Argentine's future had been thrown into question by Cortese's exit, but he reiterated his commitment to the club at a press conference on Thursday.
"I am fully committed to these players, to the staff and to these supporters. It would make no sense to leave this path," he said.
Pochettino claimed last season that if Cortese ever left the club, he too would head for the exit door.
However, he now feels the situation has changed, revealing he is not willing to leave midway through a project.
"Firstly, I want to show my gratitude in Nicola Cortese," Pochettino said. "He is a person who believed in me and brought me here."
"Eight months ago, I was at the end of my first season," he continued. "The situation now is completely different.
"We are in the middle of our project here at Southampton."
Southampton currently sit ninth in the Premier League, but have won just two of their last 10 fixtures.
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