Morris established himself as the League Two side's first choice in the second half of last season and has been handed the fresh terms.
O's manager Martin Ling had no qualms in giving the 20-year-old the contract extension and is hoping for Morris to continue his progress.
"Glenn was in superb form last year and this is a justifiable reward for him," Ling told the club's official website.
"There isn't a better young goalkeeper in divisions two or three and this is a massive period in his career.
"He's been given the number one shirt for this season but he needs to stay consistent to hold off the challenge of Lee Harrison, who will push him hard all the way."
