Cortese was said to be considering his position due to frustrations over the backing he receives.
However, the pair signed off on the plans for the club next season, meaning the Italian will remain with the south coast club.
Cortese said on the club's official website: "We are now in a position to move forward with our ambitious plans for the club and I would like to thank Katharina for the very helpful manner in which she approached our talks about long-term strategy. I am pleased that this matter is now resolved.
"We can now focus 100 per cent on our preparations for next season. We have tremendous momentum now at the club and we are looking forward with great confidence to an exciting and successful campaign next season.
"The last few days have shown me the unity that we now have across the club in the players, the first-team manager, the staff and the chairmanship which normally takes years to achieve. I have always believed that to create this unity would make us a very strong club and it is a unity of which I am unashamedly proud."
Cortese, who has overseen Saints' rise from npower League One to the top flight, was linked with a move away from the club and earlier in the week when Pochettino was asked if he would stay should the Italian leave, the Argentinean was blunt.
"Of course not, because the person that has put his faith in me in this club and as a manager is Nicola Cortese. Nicola Cortese has basically been the reality at Southampton for the past four years," he said.
"Of course, the people that are in the club, that are part of the club, that are behind the scenes in the club were a little bit worried about what has been said. Me on a personal sense, I would not understand a Southampton without Nicola being here."
Cortese replaced Nigel Adkins with Pochettino four months ago, a decision which has paid off with the Saints having secured Premier League status for next season.