The 21-year-old made his debut for Charlton as a substitute in the final game of last season.
"Good, young British goalkeepers are few and far between," Charlton boss Chris Powell told the club's official website.
"We feel we've got one who will develop into a very good goalkeeper, who will have a good career in the game.
"It's pleasing that he's agreed to sign the new contract and that he sees his immediate future here.
"Goalkeepers mature late in their careers but he's already showing signs that we've got a real talent on our hands.
"It's pleasing that for the next three years he sees his future with us."
Pope added: "It's a big club so to have a long-term future secure, for me, as a player developing, it's something you look for.
"It's about just concentrating on my work now and continuing my progression."