The move was finalised minutes before Thursday's 5pm deadline with United confirming it is a free transfer.
The 21-year-old has been on loan at Layer Road this season but Blues will be entitled to receive a percentage of any future transfer fee involving the forward.
Meanwhile, City managing director Karren Brady has revealed the Midlanders are confident of landing their own on-loan star Mikael Forssell.
Although it has been revealed this week that the Finn has plenty of Premiership suitors, Brady feels certain the Chelsea striker can be persuaded to commit his future to St Andrews.
"We see our future next season being pinned around Mikael," she said. "It was interesting for Mikael to be taken through a list of players we are hoping to be signing.
"Mikael is very happy here. I would hope that, if he is going to leave Chelsea permanently, he would come to us.
"I have no indication that is not going to happen."
