The 37-year-old won the FA Cup twice in a five-year career with the Gunners and succeeds fellow Spaniard Unai Emery, who was sacked in November.
Arteta had been working as a coach under Pep Guardiola at Manchester City, joining the club in 2016 after retiring as a player.
"We need to be competing for the top trophies in the game," Arteta said.
"That's been made very clear to me in my discussions with [owners] Stan and Josh Kroenke and the senior people from the club."