The European champions ensured the three points at the Santiago Bernabeu with a 20th-minute goal from Dani Carvajal and a Gareth Bale strike on 51 minutes.
Keylor Navas held on to his place as Real Madrid’s starting goalkeeper despite the arrival of Thibaut Courtois.
Costa Rica international Navas has been first choice since 2015 but had been expected to make way for Courtois, a recent 35-million-euro recruit from Chelsea who was voted best goalkeeper at this summer’s World Cup.
Julen Lopetegui, who replaced Zidane as coach, also selected Nacho at the heart of the defence rather than French World Cup winner Raphael Varane.
Real, playing their first La Liga game since the departure of Ronaldo to Juventus, went into Sunday’s game on the back of a 4-2 defeat to Atletico Madrid in the UEFA Supercup in midweek.
But they were rarely in trouble on Sunday.
Bale was unlucky when a diving header came back off the crossbar in the first half before the Welshman was key in the build-up to the first goal with Carvajal heading in his cross.
Another defensive error helped Real grab their second goal when Marco Asensio stole the ball and sent over a cross for Bale to score with his left foot.
Asensio was close to getting his name on the scoresheet but was denied by the woodwork.
© Agence France-Presse (additional edits from Cillian Cunningham)