Setford, who is an England youth international, arrives at the Emirates on a four-year deal, with the Gunners holding the option of a further 12 months.
Setford has been signed for the academy but will regularly train with the first team and will fly out to join Mikel Arteta's squad for their pre-season tour of the United States.
Setford is the second teenage goalkeeper Arsenal have signed this transfer window, having already brought in Lucas Nygaard from Danish club FC Nordsjaelland.
"I'm looking forward to learning new things next season and pushing myself to another level. For the short term, it's very important now to get a feel for the group and start to know everyone."
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