Mancini, who was sacked by Manchester City in May, replaces Fatih Terim after the Turkey national team manager was relieved of his duties last week.
The Italian takes charge at the Turk Telecom Arena with Galatasaray in mid-table after claiming just one win from five games in the Super Lig so far this season to leave them eight points adrift of rivals and leaders Fenerbahce.
Galatasaray opened their UEFA Champions League campaign in similarly dismal fashion, losing 6-1 to Real Madrid in their first encounter.
Mancini - who guided City to one Premier League title and one FA Cup crown during his time at the Etihad Stadium - will hope to help his side bounce back from that defeat, with his first game in charge coming at Juventus on Wednesday.
A statement on the club's official website read: "Galatasaray Sports Club and Roberto Mancini have signed a contract for three years starting from the 2013-2014 football season."
The defending Turkish champions become the fifth club Mancini has managed during his career, with spells at Fiorentina, Lazio and Inter preceding his stint at City.
Mancini enjoyed great success at Inter, winning the Serie A title three times between 2005 and 2008.
He then went on to lead City to their first league crown in 44 years in the 2011-12 campaign, but was dismissed after failing to replicate that feat the following term.
