A Jamie Smith double in the first 10 minutes of the second period lit up a sold-out Pittodrie and a Mikaal Antonsson own goal and Richard Foster's late strike emphatically secured Aberdeen's progression.
It was Aberdeen's first win in the tournament but it took them above Copenhagen into third place and secured European football beyond Christmas for the first time in more than 20 years.
Calderwood said: "I said before it was the biggest game of my managerial career. We all know the history of this great club and the lads have made their own bit.
"Some people were a bit derisory about the UEFA Cup but there were 20,000 here that definitely weren't.
"I'm so happy for the players because most of them have been together for three years and worked so hard to get here and deserved it.
The Dons boss expected a tight first half but told them to "keep believing" and Smith turned the game with a superb 25-yard strike on his return from a hamstring injury.
"It was a wonderful goal from Jamie Smith," Calderwood said. "Copenhagen had to open up after that which doesn't suit them and we punished them.
"He is so special, he has two great feet. I was a little bit annoyed with him in the first half, he had the chance to take a defender on in the box and he didn't. But I will let him off!"