The former Spain midfielder was in action just 24 hours after his move from Real Madrid went through, helping Bayern to a 1-1 draw at Schalke on Saturday.
With his first few days with the club now behind him the 32-year-old reflected on a shock move which he says is just right for him.
"I was so excited about it all," he said at a press conference. "It's an honour and a privilege to be here in Munich.
"As soon as the clubs reached an agreement, I jumped into a plane. I was surprised I made my debut so soon, but I think it was a decent start."
Alonso becomes the fifth Spanish player on Bayern's books, together with their Iberian coach Pep Guardiola.
That, according to the former Liverpool midfielder, is only one of the signs of how big the Bundesliga is becoming.
"I get the impression that more and more football fans in Spain are interested in the Bundesliga," Alonso said.
"The league's getting stronger and it's fantastic to be a part of it.
"I know I've got to get used to the pressure, but I thrive on it. Saturday was fantastic."
He may only have spent one full day in Munich following his signing, but his first impressions of the city have been positive, while the prospect of a Bavarian winter does not worry him either.
"Munich's a great city," he said. "It's very lively and there's always something to do.
"I'd only been here as a tourist before and now I'm looking forward to getting to know in depth the traditions and the culture.
"I'm Basque so I'm used to the weather too. My wife and my children are going to benefit from this move."
Alonso has also arrived with ambitions. Having already won the Champions League with Liverpool and Real Madrid, he would like to get his hands on the trophy again with a third club.
"That would be a dream," he said. "I've had a great career so far and Bayern is a new experience for me now, and it feels good.
"I like Pep Guardiola's style a lot and I think that I and the club can benefit greatly from him."