Sevilla goalkeeper Bono guided Vinicius Junior's effort into his own net in the second half, allowing Zidane's side to take the three points and move up to third in the table.
Following two defeats in the past three games, including a 2-0 defeat to Shakhtar Donetsk in midweek that leaves Real's Champions League qualification hopes in the balance, Spanish press reports have linked former Tottenham boss Mauricio Pochettino and ex-Madrid forward Raul to Zidane's job.
Zidane said his side could be satisfied with their narrow win after a "difficult" time: "These are three very important points for us. It's a well-deserved win, against a very good opponent, who can put you in trouble at any time. We got through this.
"It has been a bit difficult lately, today we have to enjoy it, be happy, but nothing more," he added.
Zidane made two changes from the debacle in Ukraine with Casemiro and Vinicius replacing Martin Odegaard and Marco Asensio.
Ex-Madrid boss Julen Lopetegui brought back Bono into the starting lineup after he missed four games with coronavirus which included Wednesday's heavy European defeat to Chelsea.
Morocco's Bono made a shaky start as his attempted clearance inside his own box was blocked by Vinicius.
He made up for the error on his own line despite Karim Benzema's efforts.
The former Girona keeper kept the sides equal at the break with a tidy save from Benzema's 38th-minute shot, but his crucial mistake came 10 minutes into the second half.
Ferland Mendy played a low cross into the area before Vinicius slid in to guide it towards the goal.
Bono came out to close the space but deflected Vinicius' limp attempt into his own goal as the ball spilled off his arms.
The hosts' Suso, making his first appearance since late October, and Lucas Ocampos had late chances to equalise but Zidane's men held on for a morale-boosting win before Wednesday's crucial Champions League group game against Borussia Moenchengladbach.
Atletico Madrid, unbeaten since the start of the Liga season, moved to the top of the table with a 2-0 win over Real Valladolid.
France forward Thomas Lemar and Spain midfielder Marcos Llorente scored for Diego Simeone's men, who are among five clubs to remain without a domestic defeat in Europe's main five leagues.
Lionel Messi's Barcelona head to Cadiz in the day's final game while Real Sociedad, who have lost just once in the league this term but now trail Atletico by two points, travel to Deportivo Alaves on Sunday.