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.