Corluka, 21, arrives from Dinamo Zagreb for an undisclosed fee, although reports from Croatia suggest City have paid about £7million.
Corluka was part of the Croatia side that defeated England 2-0 last season. He can play either in central defence or at right back.
Garrido, 22, has agreed a four-year deal after Eriksson paid the Spanish club £1.5million for the player's services.
He was looking for a new challenge after Sociedad's relegation from the top flight.
Eriksson's has now made six new signings since he took over with Rolando Bianchi, Martin Petrov, Gelson Fernandes and Geovanni all moving to the club.