Adkins was relieved of his managerial duties nine days ago after taking the club to back-to-back promotions and 15th place in the Premier League, with his dismissal arriving two days after a 2-2 draw at Chelsea.
He was immediately replaced by Argentinean Mauricio Pochettino, with the general consensus being that Adkins had been harshly treated.
The Southampton fans certainly felt so and sang their former manager`s name during the 0-0 draw with Everton last Monday, but Cortese stands by the move he made.
In an interview with Sky Sports News, he said: "Nigel and I, we always had a very good relationship, to the last second. It was very tough, not just for him but for me.
"To me, there`s one time, which is right, especially if you are 100% convinced it is the right thing. Ultimately for the team it was the right thing, it was perfect timing
"It sounds harsh but it was for the benefit of the progress we want to achieve over the next couple of months and years. It`s not only for today but for tomorrow."