Los Blancos completed a 4-1 aggregate victory over the Italians with a 3-1 win in Madrid to reach the quarter-finals for the first time since 2017-18.
Zidane's side, who are third in LaLiga and six points behind leaders Atletico Madrid with 11 games to play, will find out their last-eight opponents at Friday's draw.
Madrid won three Champions League titles in a row from 2016 to 2018, although they only won LaLiga once during that hat-trick.
“That's what we're going to try to do," Zidane said when asked if they could win LaLiga and the Champions League.
"Why not? That's what we're here for. We're going to try to do it. It's a big ask, but we'll give it our all.
"We're alive in both competitions and tonight we have to enjoy this, not just because of the goals we scored but particularly because of our performance.
"When you work on things with the players and they then come off in a game it makes you really happy, particularly when it's against a team that demands so much of you."
Croatian midfielder Luka Modric was named man of the match against Atalanta, setting up the opening goal for Karim Benzema and pulling the strings.
"He's so intelligent, that's clear," Zidane said. "He's 35 years old but you wouldn't know it out on the pitch. If Luka wants to become a coach, he's got that in him.
"He understands football, if he wants to be a coach he won't have any problem. I love the kind of guy he is.
"Today, playing the double pivot alongside Toni [Kroos], they have such great chemistry and that's fantastic for the team."