The 15-year-old will link up with Arsene Wenger's youngsters after the two club's agreed a compensation fee and he passed a medical.
Jamie, who can played in both defence and midfield, has been training with Arsenal since Monday and is set to sign schoolboy forms on Friday.
Cheltenham's head of youth Ken Oram told the club's official website: "We are delighted that Jamie Edge has accepted the offer of joining Arsenal, which is a fantastic opportunity for him to take his development to the next step.
"He has worked very hard with us and fully deserves the chance of being able to maximise the potential he has shown.
"Arsenal will make him a better player given the infrastructure and coaching environment they have there."