He 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 to get the chance to play in such an important league, the most watched in the world and where the best football is being played.
"I am also very happy to have the chance to arrive at a team that made a very big effort to give me this chance.
"I have signed a very good contract, at this league's level. All the parts involved are satisfied. I have signed a three-year deal.
"I feel that all that I am getting in my career is thanks to a lot of work and effort. I am arriving at a team that has been watching me for a long time and I am very happy for that."