The Welshman netted with his head, right foot and left foot in Saturday's 4-0 win over Real Valladolid, with the injured Cristiano Ronaldo watching on.
It was a performance which would have left the Bernabeu faithful in no doubt his world record signing from Tottenham was money well spent, with sports newspaper Marca saluting his "perfect hat-trick".
He also set up the other goal for Karim Benzema with a super cross.
Bale said on the club website: "My team-mates congratulated me for the match. It's always a pleasure to score, but even more so when you can score a hat-trick in the Santiago Bernabeu.
"I'm very happy here. I needed time to adapt after not having a pre-season, but I'm improving now. Everything I hoped for has come true. The fans and the club are incredible.
"I scored three goals but I hope to do better. I can do it. I want to keep improving and working hard during the matches and training sessions and learn from the best, like Cristiano Ronaldo. He's the best player in the world and he deserves the Ballon d'Or".
On the progress of his Spanish, Bale said: "I'm slowly learning my first foreign language but I'm not ready to use it yet. I'm working on it a lot off the pitch."
Madrid boss Carlo Ancelotti claimed Bale, who now has nine goals for the capital club, has now settled in his new surroundings.
The Italian said: "I think his period of adaptation is over. He is doing well, he is confident and is very important for us. He did very well."
Bale's display helped make light of Ronaldo's absence with a hamstring injury.
But he was left in no doubt that the Portuguese was still top dog at the Bernabeu.
Full-back Marcelo said: "There is no need to compare them, Cristiano is above the rest of the players and everyone in the team knows that.
"Bale is a great player and is doing well."