The Republic of Ireland international passed a medical on Tuesday morning and put pen-to-paper on his four-year contract at lunchtime.
The signing of the deal is the culmination of a drawn out transfer saga, with Bruce finally landing the forward from Crystal Palace.
He said: "He has been my number one target all along and I am sure he will be successful and score goals for Birmingham City."
Bruce claims the pressure is all on him rather than the marksman.
The former Palace manager added: "There's no pressure on Clinton because I'm the manager who went out to buy him, it's my judgment on the line.
"It's taken longer than expected but the club is delighted to have signed Clinton on a four-year deal."
Andrew Johnson has swapped St Andrews for Selhurst Park as part of the deal, and Blues managing director Karren Brady had words of support for the forward.
Brady said: "AJ was the first player to progress through the club's Academy and has performed really well in the first team.
"We would like to wish him all the best for his future at Palace."
Brady revealed that there could be further signings in the near future, saying: "We have been able to purchase every player on Steve's wish list to date.
"We are hopefully not finished with signings. We know that Steve wants one, if not two more and we are going to try and achieve that."
Morrison was always confident he would make the move to St Andrews, saying: "I knew deep down eventually it would happen. I am just happy I am here."
Commenting on his relationship with Bruce, Morrison added: "He is one of the best managers I have worked with, he helped me a lot during his short time at Palace and I hope he can do so again."
The £4.5 million price tag is unlikely to affect the forward, who added: "I am not nervous, I did not set the price and am confident I will score goals. I am just going to enjoy it, it is a dream for any kid (to play in The Premiership)."
