Bramall has previously undergone trials at Crystal Palace and Sheffield Wednesday without securing an offer, and he admits that he was "gobsmacked" by Arsenal's interest.
"Incredible, incredible. It's a dream come true. I can't believe that I'm here already," he told the club's official website.
"I played a game at Birmingham City on a Monday. Then on the Tuesday, I was made redundant from Bentley Motors and on the Wednesday, Arsenal came in.
"I was gobsmacked and took the chance with both hands. I drove down on the Wednesday and trained with the first team on Thursday and Friday. Then they offered me a deal."
Bramall will initially join up with the club's Under-23 squad.