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.''
Cortese has attracted criticism in the wake of the sacking, with Saints icon Matt Le Tissier accusing him of having an ego problem.
According to Cortese, though, everything he does is for the benefit of Southampton.
He added: ''Maybe I need to sacrifice my popularity to get the right decision. If that's the case, I'm happy.
''I am a perfectionist. Some people say that's not good, but it helps the football club to keep attention to detail and not stand still.''
On Le Tissier, a one-club man afforded hero status by the Saints fans, he said: ''I don't even know him, he's never met me. Anyone in the past who is associated with the club is stirring negativity towards me should probably ask if it's beneficial for the club. I would not allow myself to judge someone I have never met.''
On the future of the club, but as a whole and with Le Tissier, Cortese asked everyone to get behind his vision.
He said: ''I'd like in the future, for the supporters to buy more into what we do as a club. Not players, managers or chairmen, the club.
''I am not excluding anyone. Anyone who has Southampton at heart has a place.''
• Le Tissier wants clear-the-air talks with Cortese. Read the full story here.