The Gunners have agreed a deal worth just over £1million for the 25-year-old, who moves to the Emirates on a four-year contract.
Runarsson completed a medical in north London last week and arrives to help fill the void left by Emiliano Martinez.
Martinez shone in the Arsenal first team following the Premier League restart last season with Bernd Leno sidelined through injury.
However, with the German reinstated as Mikel Arteta's first-choice option this term, Martinez sought a fresh challenge, joining Aston Villa where he is expected to become Dean Smith's regular no1.
Runarsson - who will wear the no13 shirt at Arsenal - could be in line to make his Gunners debut in the Carabao Cup third-round clash against Leicester City at the King Power Stadium on Wednesday night.
"I'm just extremely happy, extremely proud, it is a big day for me and my family. I can't wait to get started!” he told Arsenal's official website.
“It is one of the biggest clubs in the world - they've won the Premier League 13 times, the FA Cup 14 times, it is a huge club and obviously playing in the Premier League as well, it is a win-win situation.
“I know it is going to be a lot of hard work, but I am ready to do everything I can to get the most amount of minutes as possible.”
Arsenal's technical director Edu added: "We’re very pleased to welcome Alex to our squad. We have been monitoring him for some time and from the analysis, he has strong attributes that we look for in a goalkeeper and as a person.”