The 25-year-old Iceland international joined the Ligue 1 club in July 2018. He made 13 appearances in all competitions last season.
Gunners boss Mikel Arteta said on the club website: “We want to create healthy competition for places. We look forward to seeing Alex bring further depth to the goalkeeping position.”
The deal is reported to be worth around £1million and will see Runarsson provide cover for Bernd Leno.
The north London club needed a new stopper after allowing Emiliano Martinez to join Aston Villa.
Runarsson said on the Arsenal website: “I’m just extremely happy, extremely proud, it is a big day for me and my family. I can’t wait to get started.