The decision to dismiss Adkins so soon after Saints fought back from two goals down to draw 2-2 with the European champions and at a time when they seemed to have turned a corner in their bid for Premier League survival has been widely questioned.
"Nigel and I, we always had a very good relationship, to the last second," Cortese told Sky Sports News. "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 percent convinced it is the right thing. Ultimately for the team it was the right thing, it was perfect timing."
Among those who raised their eyebrows when Adkins was replaced by Argentine Mauricio Pochettino on Jan. 18 was Manchester United manager Alex Ferguson, who said it was a "crazy world".
When Adkins was shown the door, Southampton had lost just twice in their last 12 Premier League games.
"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," added Cortese, who built his reputation in Swiss finance.
Fans, disgusted by the sacking of the man who had led the south coast club to back-to-back promotions, had threatened to protest during Pochettino's first match in charge although that did not materialise in a 0-0 draw with Everton on Monday.
"Maybe I need to sacrifice my popularity to get the right decision. If that's the case, I'm happy," Cortese said.
"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."