An own goal by Sevilla keeper Yassine Bounou secured all three points for Los Blancos.
Madrid, who rise to third in the La Liga table, are four points away from league leaders Real Sociedad, while Julen Lopetegui's Sevilla are fifth in the division.
Zidane went into the match under pressure following two successive defeats, to Shakhtar Donestk in the Champions League and Sevilla in La Liga.
Zidane's side almost took the lead inside a minute after Vinicius Jr. was played through on goal by Rodrygo, but the winger shot wide from a tight angle.
Sevilla missed their chance to go ahead in the 33rd minute after Jesus Navas's cross found the head of defender Jules Kounde, but the defender missed the target and couldn't guide it past goalkeeper Thibaut Courtois.
But after the break Vinicius Jr.'s shot was turned in by Bounou to send Real into the lead.
Bounou is the first goalkeeper to score an own goal in La Liga this season and the first Sevilla goalkeeper to score an own goal in the competition in the 21st Century.
After the match Vinicius Jr. claimed that the goal should be awarded to him and not for Bounou.
"It's mine, of course! I took the shot," he said. "The win today is really important for our confidence.
"It's a difficult place to play against a good team. We had some games where we played well but didn't win, and today we won... We didn't have any doubts today."
Meanwhile, manager Zidane said that his team deserved to win and praised the performance.
"They're three really important points. But beyond that, it's the game we played, from the first minute to the last. We suffered a bit in the second half but our first half was really good. We deserved it, against a really good opponent who can cause you problems," he said.
"It's important for the next game. It's been a complicated moment lately, today we have to enjoy it and be happy. They're three points, now we have to rest well and think about what comes next."
Zidane spoke about the criticism both he and his team have received despite claiming victory against Sevilla.
"It's like that. It won't change much. When you lose, there'll be criticism. Today we have to think positive. We have to keep looking for consistency. The rest won't change."