Bale opened the scoring with a powerful header in the second minute and rounded off the win four minutes into added time with a simple finish after new arrival Marco Asensio had scored his first league goal for Real in the 40th with a sumptuous lob.
Elsewhere, title rivals Atletico Madrid were denied a win with virtually the last kick of the game and had to settle for a 1-1 draw at home to promoted Alaves.
The game looked to be heading for a goalless draw until the home side were awarded a penalty in injury time for a handball, allowing Kevin Gameiro to mark his Atletico debut by converting from the spot in the 93rd minute.
Diego Simeone's side were dealt a nasty late surprise though when Manu Garcia let fly from outside the area to salvage a point for the Basque side, the 2001 UEFA Cup runners-up who were playing their first top flight game in 13 years.
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Earlier on Sunday, Sporting Gijon beat Athletic Bilbao 2-1 thanks to quick fire goals from Duje Cop and Victor Rodriguez at the start of the second half, with Borja Viguera scoring late for the visitors.
RACIST CHANTS
Midway through the first half referee Carlos Clos Gomez halted the game for a minute after Athletic striker Inaki Williams, a Spain under-21 international born in Spain to a Ghanaian father and Liberian mother, was subjected to racist chanting from the home supporters.
Real are third in the table with three points, behind Barcelona and Sevilla on goal difference, while Atletico are seventh.
After title rivals Barcelona had laid down a marker with a ruthless 6-2 win at home to Real Betis on Saturday, the pressure was on Real to make a strong start.
They were without first choice forwards Cristiano Ronaldo and Karim Benzema, so Bale was accompanied by Alvaro Morata and Asensio instead.
The Wales international had nodded in the only goal in this fixture last season late in the second half but needed little more than 60 seconds to strike this time, meeting Dani Carvajal's whipped cross from the right to power the ball into the top near corner.
"It's important to get off to a winning start especially at a ground as difficult as this one. This was the best way to start the season," Zidane told reporters.
"Every player put in a huge effort and we attacked and defended together. If we fight for every ball we can achieve big things."
Real's second goal came in the 40th minute via Raphael Varane's scooped pass from defence up to Asensio, the youngster controlling with one touch before producing a delightful chip over Geronimo Rulli.
Bale completed the win when he latched on to James Rodriguez's through ball to race clear and calmly take it round Rulli to score again.
(Editing by Alison Williams/Rex Gowar)
