Hajrovic moves to the Turk Telekom Arena on a four-and-a-half-year contract after agreeing to leave the club he first joined in 2000.
The 22-year-old broke into the first team at Grasshopper in 2010 and has since established himself as a regular for the Swiss Super League side.
The attacking midfielder has helped move Grasshopper into fourth place in the Super League, scoring six goals in 15 league appearances this season as Michael Skibbe's men aim to win their first Swiss title in 11 years.
However, Hajrovic will now turn his attentions to Galatasaray, who sit eight points behind fierce rivals Fenerbahce in the Turkish Super Lig title race.
Hajrovic could make his competitive debut for Roberto Mancini's side when Galatasaray host Tokatspor in the Turkish Cup on January 15.
