Cortese dismissed Adkins earlier this month just two days after Southampton had come from 2-0 down to grab a draw against European champions Chelsea.
Adkins, who led the Saints from League One and back into the Premier League, was immediately replaced at St Mary's by former Espanyol coach Mauricio Pochettino.
Saints fans have voiced their belief that Adkins has been harshly treated, but Cortese insists he took the decision for the benefit of the club's long-term future.
"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% 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, but he stressed that 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."
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