The Holland international completed his medical with the Turkish giants earlier on Tuesday before signing his contract at a ceremony at the club's Turk Telekom Arena.
The 28-year-old, whose transfer fee from Inter is believed to be in the region of 10million euros (£8.4million), has penned a reported three-and-a-half-year deal with the Super Lig leaders.
Sneijder's move ends a three-and-a-half-year spell at Inter during which he won a famous Champions League, Serie A and Coppa Italia treble in 2010 under Jose Mourinho.
The playmaker, who joined Inter from Real Madrid for 15million euros (£12.6m) in 2009, had not featured for the Nerazzurri since a 2-0 victory at Chievo in September after publicly refusing to take a 2million euros (£1.7m) pay cut on a contract that ran until June 2015.
Inter head coach Andrea Stramaccioni maintained the playmaker's absence from the first team was solely a tactical decision, despite Inter's clear difficulty in keeping pace with league leaders Juventus through an injury crisis.
Sneijder said: "I would like to thank Galatasaray for the transfer.
"It is a great honour to be here. I saw thousands of fans waiting for me. It was incredible, the last two days I've had a very special feeling.
"This is a great stadium. I'm ready, I'm happy, I'm excited. And I want to get
started right away. I want to say again, I am very happy."