Karim Benzema netted his 100th league goal for the club with an early opener in Real's first game since the departure of Rafael Benitez.
Bale headed the second, hit the bottom corner after the break and wrapped up his hat-trick with another header.
Benzema scored the fifth to leave Real two points behind leaders Barcelona.
Barca's comfortable 4-0 victory over Granada put added pressure on Real at the outset, but they responded in style under their new boss.
Did Zidane change much?
Not really. When you take over a side that has lost one of its previous nine games, scoring 36 goals in the process, there is not much need to.
The 43-year-old gave a low-key acknowledgement to the rapturous welcome he was given from supporters - a couple of modest waves hinting at the Frenchman's desire not to be the centre of attention.
On this occasion, his players did all the talking.
The inclusion of the attack-minded Dani Carvajal in defence as one of two changes suggested Zidane wants his full-backs to get forward more. He was repaid when the right-back, in for Danilo, set up the first of Bale's goals.
It was the kind of line-up the fans wanted to see.
Bale makes the perfect impression
As sporting director of Los Blancos, Zidane was instrumental in Bale's record-breaking £85.3m move to the Bernabeu from Tottenham in 2013. Should he continue to produce displays like this one, the Welshman will continue to be a centrepiece of his new manager's plans.
Bale did not start too well, and he should have been flagged offside as Benzema claimed Real's first goal.
But, from that point, the 26-year-old's contribution was wholly positive and he was cheered from the field when replaced by Isco with 16 minutes remaining.
What does Zidane need to address?
There remain defensive frailties which would have been exploited by a better side than Deportivo.
The visitors should have taken the lead, and Zidane will have been alarmed at the ease with which his side allowed crosses into their area, especially in the first half. A lack of communication between his centre-backs also needs to be addressed.
Of course, any side possessing Cristiano Ronaldo, Bale and Benzema will score goals but, despite the clean sheet, Real continue to look vulnerable.
What Zidane said
"Whenever you win a match, then the coach can only be happy, and I'm happy," said the Frenchman. "What I liked most of all was the attitude of everyone, those who played and those who didn't.
"Winning 5-0 isn't easy against a side like Deportivo and I'm happy with our first game and our first four days of training."
What next?
Real have more than a week to prepare for their next game, when they host struggling Sporting Gijon on 17 January. It's a quicker turnaround for Deportivo, who welcome second-tier side Mirandes in the Copa del Rey on Tuesday, following a 1-1 draw in last week's first leg of the last-16 tie.
LINEUP, BOOKINGS (3) & SUBSTITUTIONS (6)
Real Madrid
- 01 Navas
- 15 Carvajal
- 03 Pepe
- 04 Ramos (Varane - 45' )
- 12 Marcelo
- 08 Kroos
- 19 Modric
- 11 Bale (Jesé - 74' )
- 22 Isco (Rodríguez - 66' )
- 07 Ronaldo
- 09 Benzema
Substitutes
- 02 Varane
- 10 Rodríguez
- 13 Casilla
- 14 Casemiro
- 18 Vázquez
- 20 Jesé
- 23 Danilo
Deportivo de La Coruña
- 01 Lux Booked
- 11 Juanfran
- 14 Arribas Booked
- 12 da Silva Junior
- 03 Navarro Corbacho (Correia Pinto - 66' )
- 04 Bergantiños García
- 05 Mosquera Booked
- 17 Nicolás Cartabia (Gutiérrez - 45' )
- 21 Luis Alberto
- 19 Fajr (Riera Magem - 66' )
- 07 Pérez Martínez
Substitutes
- 09 Riera Magem
- 10 Domínguez Lamas
- 15 Sanabria Ruiz
- 16 Correia Pinto
- 23 Lopo
- 24 Gutiérrez
- 25 Fernández Muñiz
Ref: Pedro Jesús Pérez Montero
Att: 71,267
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