Wilshere, who turned 17 on New Year's Day, became the youngest player to represent the Gunners in league football when he came off the bench against Blackburn in September.
Manager Arsene Wenger said: "We are delighted Jack has signed a professional contract with us.
"He is a player with great potential, with an ability to find the final ball and also score goals.
"Jack is a passionate and committed young player, he is not afraid of tackles and I am convinced he will have tremendous penetrative power in a few years.
"We have already seen his qualities in both his first team performances and in training on a regular basis, and we look forward to Jack being an important part of our first team for years to come."