Pressure was on Madrid following their 2-1 defeat at home to Manchester City in the opening leg of the Champions League last-16 tie on Wednesday.
Madrid conceded twice in five minutes during the final 12 minutes in a stunning finish against City, however, Zidane's side bounced back with a 2-0 victory over Barca on Sunday.
Vinicius Junior and substitute Mariano Diaz scored in the second half at the Santiago Bernabeu as Madrid leapfrogged defending champions Barca atop the LaLiga table.
Zidane expressed his pride post-match, telling reporters: "The reaction today has been important. The dressing room is not touched. It was a complicated week, but we prepared this game well. That means the players are in and want.
"We played 78 very good minutes and in the end we lost. Today we win playing well from start to finish. We need to stick to what we believe, but we know that it will be hard until the very end. We need to prove ourselves every time we play."
"We deserved the victory because of everything we did both defensively and offensively," Zidane added. "The first half was even and our press up front was not perfect. In the second half we improved our press and we were better up top.
"We really pushed them hard and almost overwhelmed them. These are three important points because it has been a difficult week and today we had the opportunity to change that. It's three points and we regained the lead, but we're going to have to fight to the end.
"We're happy because we've all shown that we've done very well and I'm proud of them because it's not easy to win like this. Our strength today was very important. The first thing is to defend well together and then we know that at any moment we can have opportunities to score goals.
"The second half was ours. We had a lot of chances to score and we netted twice. We kept a clean sheet and [Thibaut] Courtois was instrumental in that. We won the game as a team today."
Vinicius – at the age of 19 years and 233 days – became the youngest Clasico goalscorer this century, surpassing Barca superstar Lionel Messi with his 71st-minute opener before Mariano completed the win in stoppage time.
"Vinicius was good and he scored an important goal in an important game. I'm happy for him. He was good offensively and defensively he did a great job helping Marcelo with [Nelson] Semedo," Zidane said.
"He has conditions, but he is very young and sometimes the criticism is difficult. He has to live with that because he is at Real Madrid."