Cotter, who succeeded in his first season with the side after so many others failed, said having a longer preseason than usual because of last year's Rugby World Cup helped mould the style he felt the team could play.
Then, when the All Blacks joined them after their time off, it was about getting them to understand the changes.
He said by the time they went to Japan for their preseason games, what he wanted was taking shape, and it grew from there.
"They picked it up and ran with it. The key was giving them the opportunity to do that. It's not driven.
"The boys picked it up and said, 'Yeah, we'll go with that, and they drove it themselves."
Captain Patrick Tuipulotu, who defied the odds to get on the field, said that with two minutes to go, he was sitting on the substitutes' bench with a smile, knowing they had finally won the title.
"It means so much."
Cotter said the night was perfect for the side.
"Even the rain turned up for us. The team's forwards played well, and it's been building since we came together.
"It's been our objective right from the start and the boys haven't faltered. There were a couple of losses on the way, but they've been focused on getting here. It's been 21 years, a long time between drinks, as we know."
Cotter said Tuipulotu, playing two weeks after he was expected to be out of action with a medial ligament injury, was inspirational.
"His carries, his defence, his lineout and just his presence. It was going to be a battle of the tight fives, so he was instrumental in getting us going forward, keeping us going forward and dragging people with him.
"That's what you want from a leader and he did a great job."
When he was relieved of duty and substituted, he walked off to a standing ovation. Then flanker Dalton Papali'i took over, and when he was substituted, centre Rieko Ioane led.
"This is part of the team that's been so pleasing, it's been everybody pitching in, doing what they can to achieve the objective we started with at the beginning of the season.
"They've all grown and become better, and that's an adaptive mindset that we've been developing all year."
Tuipulotu said Cotter's ability was based on simple stuff, and while the team was full of competitors, he developed the rugged edge that had been lacking.
"He'll straight shoot you. I got that the first couple of times I met him.
"That's been a good growth for this team; keep things simple, and when things are simple we know what we're doing and can do that to the best of our abilities. It was very instrumental in how we want to play."