Karim Benzema got the Zidane era off to an ideal start by scoring after just 15 minutes and then Bale netted either side of the break before scoring a second header to complete his treble.
Benzema added a fifth in stoppage-time as Real ensured the gap to Barcelona at the top remains at two points, albeit with Los Merengues having played one game more.
Earlier in the week Zidane confirmed reports that Bale was irritated by the sacking of Rafael Benitez, but the Wales international quickly showed he can work just as well under the Frenchman with a dominant performance.
After a stop-start season under Benitez, Madrid looked liberated under Zidane, once the club's record signing and still an idol thanks to his stunning winning goal in the 2002 Champions League final.
The 43-year-old's name was cheered heartily before the game by the same supporters who had booed Benitez just 10 days earlier, and they were also delighted at the sight of Isco, who was given his first start in exactly two months. The Frenchman's only other change to the team that drew 2-2 in Benitez's last game was homegrown defender Dani Carvajal coming in for Danilo, who has been inconsistent since signing from Porto last year for 30 million euros.
Zidane announced in his press briefings that he would stick with the usual front three of Bale, Benzema and Cristiano Ronaldo, saving special praise for the Portuguese, who he called "the soul of the team".
Although Ronaldo failed to get on the scoresheet, he was much livelier than at the Mestalla during Benitez's last game in charge, taking nine shots on goal and providing the assist for Bale's second goal.
Madrid made a nervy start to the game, however, with Deportivo's Luis Alberto firing over in the opening minutes and then Faycal Fajr forced Keylor Navas to tip away his shot from wide. The goalkeeper was alert again a couple of minutes later to thwart Lucas Perez in a one-on-one, and the same doubts that pervaded Benitez's side were looking all too evident.
But just a few moments later Benzema struck at the other end to put Madrid on their way to a huge win. Depor failed to clear a corner and Sergio Ramos sent the ball back into the danger area, and Benzema helped it into the net with a flick of the back heel to score his 100th Primera Division goal. The goal angered Depor as Gareth Bale was offside, and although he did not touch the ball, he may have obstructed the view of goalkeeper German Lux.
Bale got the better of the goalkeeper just seven minutes later, heading home a cross from Carvajal from six yards out.
The former Tottenham man got his second of the evening four minutes after the break, tapping in Ronaldo's low cross, and then grabbed his second hat-trick of the season by heading in a Toni Kroos corner, netting his 50th goal for Madrid in 108 games, one more than Zidane managed in 231 games in a white shirt.
Benzema rounded off the victory with a thumping shot that bounced in off the crossbar as Zidane enjoyed a comfortable start to life in the Bernabeu dugout.