Milan claimed a 1-0 aggregate victory after a goalless draw on a freezing night in London as the hosts crashed out with a whimper.
"I like to think there’s nothing impossible in sport," Pioli said.
"Clearly, only the best in Europe remain, and each round gets more difficult, but I am sure that all the European experiences, especially the negative ones – help us step up a level.
"Now we must go into the quarter-final with confidence and belief."
Milan have turned around an abysmal start to the year and are now in fifth place in Serie A, equal on points with AS Roma who are in the fourth and last place that gives Champions League football next season.
"It is wonderful to play in the Champions League, gratifying and important for Milan. So either we win it, which would not be easy, or try to qualify by finishing in the top four in Serie A," Pioli said.
Milan have booked their spot in the last eight with Chelsea, Bayern Munich and Benfica, while the remaining four teams will be decided next week.
"We’ll be paired with a great opponent, but Tottenham are a big team too and we can always have our say. It is good to dream. Now we need to focus on Serie A, where we have some lost ground to make up."
The draw for the quarter-finals will be held on March 17.
