"Signing for the club you grew up supporting as a kid," Ryan tweeted after the announcement alongside a check mark.
"Buzzing to be beginning this new chapter and will give everything I've got to contribute to the first club I ever loved."
Ryan has 59 caps for Australia, playing in the 2014 and 2018 FIFA World Cups, as well as the 2015 and 2019 Asian Cups.
He was awarded the Best Goalkeeper of the 2015 tournament which was held in Australia.
"We know Mat very well through his performances with Brighton in recent seasons and he brings additional quality to our squad," Arteta said in a statement.
"Mat has very good Premier League experience and has played over 100 times in the league, which is something that will benefit us greatly on and off the pitch. We welcome Mat to Arsenal and look forward to working with him this season."
The club then missed out on signing David Raya from Brentford as his replacement and opted instead to bring in Alex Runarsson, who has failed to provide sufficient competition for established number one Bernd Leno.