Bojinov, who spent last season at Juventus, has agreed a four-year deal.
City chief executive Alistair Mackintosh said: "Our signing of Valeri is another sign of the impact made by Dr Thaksin Shinawatra and Sven-Goran Eriksson. Valeri is an exciting young player and a great addition to the squad."
The 21-year-old is Eriksson's fourth signing in two days after deals for Javier Garrido, Vedran Corluka and Elano were completed on Thursday.
He is Eriksson's eighth signing since being appointed City manager last month.