The Chile international signed a long-term deal to dash Liverpool's hopes of bringing him to Anfield.
The Reds had hoped to include him as part of the deal which eventually took Luis Suarez to the Nou Camp.
Sanchez said: "I'm so happy to be joining a club which has a great manager, a fantastic squad of players and huge support around the world.
"I'm looking forward to playing in the Premier League and Champions League."
Sanchez eventually scored 80 goals in 166 games for the Gunners, winning the FA Cup twice, before making the move to Manchester United in a swap for Henrikh Mkhitaryan in 2018.
It was an ill-fated spell at Old10 Trafford, where he was reportedly earning £14million a year after tax, and he scored just five times before joining current club Inter Milan, initially on loan, in 2019.