Fabianski, 29, has signed a four-year contract which will start on July 1, after his Gunners deal fully expires.
Having spent most of his career as a back-up, Fabianski insisted he had come to Swansea to be first choice.
"If I'm completely honest, the main reason I came to Swansea was because I want to be the number one goalkeeper," he said. "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."
The former Legia Warsaw shot-stopper spent seven years at Arsenal following a £2.1m switch in 2007.
His best campaign was 2010-11, when he featured in 14 league games, but he only made one appearance in the Premier League this season.
However, Fabianski was mostly used in cup competitions, with his last game for the Gunners coming in the FA Cup final victory over Hull on May 17.
Arsenal boss Arsene Wenger had said he would like to keep Fabianski, but the Pole was determined to secure first-team football and will compete for a starting place with Netherlands international Michel Vorm and his German no.2 Gerhard Tremmel.
FA Cup winners Arsenal will now be on the lookout for a no.2 goalkeeper to Szczesny. Norwich's John Ruddy, Cardiff's David Marshall and Iker Casillas are among the names linked with the North London club, with Emiliano Viviano's loan from Palermo set to end next month.
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- Wojciech Szczesny