The 28-year-old Spaniard will become a Norwich player on July 1 after he agreed terms on a two-year deal with an option for a third year, while Lazio will receive an undisclosed fee.
Garrido has been a regular in Chris Hughton's team this season, making 34 appearances in total, and the Canaries boss said there was little doubt in his mind that he was a player the club wanted to keep.
"I'm delighted to have secured the services of Javi on a permanent basis," Hughton said.
"He has played 34 games for us this season and the fact that we've signed him is a testament to the contribution he has made to the team since joining us last summer."
Garrido said: "I'm so happy to have signed full-time I am very comfortable with everything here - with the players, the technical staff and of course the fans.
"I am just very happy that we have managed to remain in the Premier League where we belong.
"Now I want to do everything I can to help Norwich continue to be successful in the Premier League. For me personally I am delighted to have secured my future playing in English football with Norwich."