The Chilean winger arrived St Andrews on transfer deadline day from Mexican side Club America in a reported £4million move.
Beausejour enjoyed a sterling World Cup for Chile, prompting City boss Alex McLeish to swoop for the 26-year-old midfielder on a three-year deal.
And the player says he is delighted to have completed a move to the Midlands outfit, citing the club's desire to secure his signature as the prevailing factor in the deal.
Beausejour told Radio Agricultura: "I am very happy and proud for getting the chance to play in such an important league, the most seen over the world and where the best football is being played.
"I am also very happy for having the chance to arrive to a team that did a very big effort for giving me this chance.
"I have signed a very good contract, at this league's level. All the parts involved are in satisfaction. I have signed a three-year deal.
"I feel that all what I am getting in my career is thanks to a lot of work and effort. I am arriving to a team that has been watching me since a long time and I am very happy for that."
Beausejour agreed terms at Birmingham on the same day McLeish sealed moves for fellow midfielder Alexander Hleb and centre-back Martin Jiranek.
The Chilean said his preferred position was a wide-left berth, although he insisted he is happy to play anywhere on the pitch.
He added: "I like to play on the left but I will do it where the coach wants me to.
"For what I have spoken with him, he wants me to play where I most like and feel comfortable, and that is on the left."