The 24-year-old has struggled for fitness since completing his world-record €100 million move from Tottenham in September, but appeared to be back to his best on his first Bernabeu start.
Bale picked out the top corner after 14 minutes to open the scoring, before netting his second via a deflected free kick, and the Wales star has vowed to perform even better as he regains fitness.
"I felt good and I played the whole match, but I have to keep getting better and work hard to be 100 per cent as soon as possible," the winger told the official Madrid website.
"It's been a great night and was important for my fitness level. We played very well and hopefully there's much more to come in the future."
The Welshman will hope for another opportunity to continue his comeback when Real Madrid make the trip to Rayo Vallecano on Saturday before facing Juventus in the Champions League three days later.
"It's a very exciting time," he added.
"We have big matches ahead and everyone's looking forward to them. We're feeling very confident and hopefully we can keep winning and giving performances like tonight's."