His side carved out plenty of chances to beat Sevilla comfortably but could not find a decisive finish, until the home side's goalkeeper ran out of luck in the second half.
Ferland Mendy's low cross looked sure to be turned home by Vinicius Junior, but he made only the slightest of connections and the ball deflected in off Bounou, with Thibaut Courtois' late save from Lucas Ocampos at the other end securing a valuable victory.
Bruised by a midweek defeat to Shakhtar Donetsk that has put Madrid in a European pickle, Zidane needed a reprieve going up against former Los Blancos boss Julen Lopetegui at Estadio Ramon Sanchez-Pizjuan.
Madrid spurned a glorious first-minute chance when Rodrygo found Vinicius unmarked in the penalty area, but a low shot towards the far corner bounced just wide.
Bounou then almost gifted Madrid an opener when his attempted clearance rattled against Vinicius and looped across goal.
Karim Benzema was waiting to nod in from close range but was beaten to the ball by defender Diego Carlos, who cushioned a header to the relieved Bounou.
Toni Kroos and Benzema went close before the break, but Sevilla's resistance was ended on the 55th-minute mark.
Benzema's clever pass picked out Mendy on the left, and his low centre should have been ideal for Vinicius but turned out to be far from it for Bounou.
His head-in-hands moment made Bounou the first Sevilla goalkeeper to score an own goal in LaLiga in the 21st century.
Suso went close to a spectacular equaliser but sent his shot just too high, with Courtois thankful to see it zip off target, before Madrid's goalkeeper saved well from Ocampos' acrobatic volley to seal the points.