No financial details were given but the transfer is reported to be worth about five million pounds ($6.04 million).
"I'm very happy with the decision I made, it was a well-thought one," Torreira told the club's TV.
"I'm looking forward to starting this new adventure.
"The most important thing is to try to get used to the city, also to get to know the club, my team mates. I want to train already."
Torreira joined Arsenal in July 2018 from Sampdoria and made nearly 90 appearances for the Premier League club.
He spent the last two seasons on loan - firstly at Atletico Madrid, helping them to their 2020-21 LaLiga title, and then with a return to Serie A by joining Fiorentina.
Galatasaray have also completed the signing of Belgium forward Dries Mertens (35) until 2023, the club added on Monday.
The Belgian has joined the Turkish side on a free transfer after nine years at Napoli where he became the club's all-time top scorer with 148 goals in all competitions.
Galatasaray start their season off against Giresunspor on Friday - follow it with Flashscore.
