Alessandro Matri and Fabio Quagliarella fired Juve to a 2-0 victory in Turin to complete a 5-0 aggregative win over the Scottish Premier League side and seal a fifth consecutive success in Europe for the hosts.
The campaign marks Conte's first outing as a manager in the Champions League and he and the players are relishing the experience.
"This is our first Champions League in many years, while it is the first for me and many of our players," Conte told Mediaset.
"The Champions League represents a wonderful dream that we earned last season by winning the Scudetto.
"We got through a really tough group, have qualified for the quarter-finals and we're very happy."
Juventus entered Wednesday's home clash boasting a 3-0 first-leg lead but Conte says his side could not sit back against the visiting Bhoys.
"In all honesty, if you ask me whether I was relaxed before kick-off, then I'd say no. It's not easy to face this type of game, even with a 3-0 first-leg lead," he said.
"I and many of the players do not have much experience of two-legged ties, so we had to take each game as a lone event. We had to be wary early on and avoid conceding free-kicks as they are so strong in the air, as a goal could've introduced some fear into the team.
"We had prepared the game to take advantage of spaces when going two against two with their defenders."
Conte was full of praise for the Celtic fans that made the trip to Turin and voiced their support all the way to the final whistle.
"I want to thank the fans for their efforts tonight and compliment the Scottish supporters," he added.
"To see them singing like that when 5-0 down on aggregate is truly something extraordinary. They deserved the applause of the whole stadium."