The 29-year-old will join the Welsh club on a free transfer on July 1 - ending seven years with the Gunners.
Fabianski, who has 21 caps for Poland, found his first-team opportunities at The Emirates limited by the emergence of international team-mate Wojciech Szczesny and turned down the offer of a contract extension.
Fabianski told Swansea's official website: "I am very happy with the move, the decision and the club. I'm very excited for the coming season and the coming years.
"It will be a big honour to represent Swansea City. Everyone interested in football knows that Swansea is a proper football club. The way they play football is nice to watch and they have always been an exciting team.
"Based on what the club has achieved in their three previous seasons in the Premier League, you can clearly see that year upon year the club is moving forward.
"The club represents good values and their style of football is something that I would like to be a part of.
"I know quite a bit about them already but I am sure I will get to know quite a bit more in the coming months. For me, it was an easy decision to make to join the club."
Fabianski is the first signing made by recently-appointed Swansea manager Garry Monk and will now provide goalkeeping competition for Dutch incumbent Michel Vorm and German veteran Gerhard Tremmel at The Liberty Stadium.
"If I'm completely honest, the main reason I came to Swansea was because I want to be the number one goalkeeper," Fabianski added. "I want to play week in, week out.
"It's going to be a real fight, but I'm not afraid of that. Competition is always good, so I think everyone will take positives from that.
"I'm excited. I've already asked for the pre-season programme to see how it looks. I'm really looking forward to starting with my new team-mates and I think it's going to be a fantastic new chapter in my life.
"I will be working hard for the club, giving everything every single day. I can promise that."
Arsenal manager Arsene Wenger spent £2m on taking Fabianski from Legia Warsaw in 2007 and seven years later he signed off his time with the north London giants with the FA Cup final victory over Hull City at Wembley on May 17.