The Welshman, who moved to Real in a world-record 100million euros (£85.3million) transfer last month, opened the scoring with a stunning curled effort and then doubled the lead from a free-kick, albeit with the help of a deflection.
Cristiano Ronaldo, fresh from his row with Sepp Blatter over comments made by the FIFA president, plundered a hat-trick while Karim Benzema ended a two-month goal drought for Los Blancos with two goals.
Bale had produced a disappointing performance during the defeat at Barcelona last weekend and subsequently came in for criticism from the Madrid media but he responded perfectly at the Bernabeu.
However, rather than bask in any glow of satisfaction, the former Tottenham player vowed to continue striving to return to the sort of form and fitness that had Real paying their world record fee.
In quotes reported on realmadrid.com, Bale said: " I felt good and I completed the whole game.
"I have to keep improving and working hard to be one hundred per cent as soon as possible.
"It was a great night and important to my fitness.
"We played very well and hopefully we will have more games like this in the future.
"Now I continue to improve and work hard and as soon as possible try to be one hundred per cent."
It was by some way his best performance for the team who are now six points behind leaders Barcelona.
Bale added: "It was a great game and a great atmosphere. We played very well and we got a lot of goals but the important thing is to have got the three points and we have scored many goals which is very positive for us.
"The downside has been that we have received several goals, but we were good, it gives us confidence.
"We have a lot of confidence and hopefully we can keep winning and producing performances like tonight."