The decision was taken after Southampton's 2-2 draw with Chelsea in the Premier League, and the fans showed their loyalty to Adkins by chanting his name throughout the team's next league outing against Everton, which they drew 0-0, Pochettino's first game in charge.
Cortese believes the timing of the Pochettino appointment was perfect for the club going forward, and feels his relationship with Nigel Adkins was never an issue.
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"To me, there's one time which is right, especially if you are 100 per cent convinced it is the right thing. Ultimately for the team it was the right thing, it was perfect. 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."
Saints legend Matt Le Tissier is just one of many who have criticised Cortese's sacking of Adkins, but the Italian banker feels the future of the club is the most important thing, and not what Le Tissier and the fans think of him.
"Maybe I need to sacrifice my popularity to get the right decision. If that's the case, I'm happy," he continued.
"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's negative comments, Cortese said, "I don't even know him, he's never met me. Anyone in the past who is associated with the club [who] 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.
"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."
Southampton currently sit 15th in the Premier League, four points above the relegation zone, but with just two wins from their last 10 league games, and with league leaders Manchester United their next opponents at Old Trafford on Wednesday.