The pressure on the head coach was eased at Mestalla as the champions claimed a hard-fought victory that takes them to within two points of third-place Valencia, who have played a game more.
Two first-half penalties from Cristiano Ronaldo set the visitors on course for their first away win in LaLiga since October and fine late goals from Marcelo and Toni Kroos made the win safe after Santi Mina had given the hosts a lifeline.
Zidane was delighted with the display, especially in the wake of Madrid's Copa del Rey exit at the hands of Leganes this week, and suggested the result will help the squad to overcome something of a mental block.
"I go away very happy," he told a news conference. "We have to congratulate the players. We played an amazing game on a pitch that's never easy, but we had to do it.
"We did really well and we earned the victory, which was important. There are difficult moments and it shows the players have personality.
"In football, a lot of things are in the head. The players know what they do, they don't give up.
"We interpreted the game very well tactically. They're a direct rival and we have a game in hand. If we win, we'll be above them.
"In the second half, Keylor [Navas] saved us for 15 or 20 minutes, but we deserved the win overall."
Zidane also shrugged off Karim Benzema's frustrated reaction to being substituted in the second half, adding: "Everyone is annoyed when they get replaced, but what matters is the result and Karim is happy."
Goalkeeper Navas, who made an important save to deny Dani Parejo with the scores at 2-1, insists Madrid will not give up the chase in LaLiga despite them being 16 points behind leaders Barcelona.
"We played a great game, which is the most important thing. We had to suffer in some moments but we didn't lose personality and then the [late] goals came," he said to beIN SPORTS.
"The team has been playing well for a lot of games. It's no secret that we've also played very badly, we know that. What we have to do is maintain consistency and we're strengthened by matches like these.
"Valencia are a great opponent, it's at their home, the fans support them and they give everything, so it's not easy. When you come to a place like that and you think they're not going to create a single chance, you're very wrong.
"We know LaLiga is very difficult but we're going to give everything. We won't throw anything away. Whoever thinks we're not going to fight in every match until the end to finish as high as possible is very wrong. We are Madrid."