The former Manchester City boss has agreed a three-year deal with the Istanbul giants, the club announced on their official website.
The Italian's first match in charge will be Wednesday's Champions League clash at Juventus.
Mancini, who was sacked as City manager in May, said in quotes reported on the official Galatasaray Twitter feed: "I am very happy to come to a club as big as Galatasaray.
"The Galatasaray-Juventus match will be a happy coincidence for me."
The Super Lig club confirmed Mancini would be paid 3.5 million euros (£2.9 million) for the current season and 4.5 million euros (£3.8 million) for the following two, plus bonuses.
Mancini takes over from Fatih Terim, who was sacked earlier this week after rejecting a contract extension.
The 48-year-old Italian had been out of work since leaving the Etihad after a three-and-a-half-year reign at the Etihad Stadium which included a Barclays Premier League title success in 2012 and 2011 FA Cup glory.
He failed to make an impact with the club in the Champions League, though.
Galatasaray - who in January spent heavily on the likes of Didier Drogba and Wesley Sneijder - have had a disappointing start to the season, taking only seven points from five league games, while they suffered a 6-1 home thrashing at the hands of Real Madrid in the Champions League.
Wednesday's match against Juventus could pitch Mancini into battle against his former striker Carlos Tevez.
When both were at City Mancini memorably claimed the Argentinian refused to play for him in a Champions League match against Bayern Munich and the forward was frozen out of the club for five months.
Tevez is, however, a doubt for the match because of an ankle problem.