The 21-year old will now join up Mikel Arteta’s side from French team Dijon, after Arsenal had a £1.8m offer accepted by the Ligue 1 club.
Arteta told an interview with the club’s official website he was delighted to final complete a deal for the highly rated stopper, with Runarsson penning a five-year deal at the Emirates.
Runarsson is expected to replace Martinez as Bernd Leno deputy this season, with Matt Macey potentially heading out on loan in the coming weeks.