"Our first half was okay, while we got under heavy pressure in the second," Copenhagen captain Lars Jacobsen said. "However, we had a fantastic goalkeeper, and that is what you need in a match like this."
Juventus pushed hard in the final minutes in search of a winner, with Arturo Vidal and substitute Sebastian Giovinco both having chances.
Juventus had traveled to Copenhagen after having played seven matches in 25 days, and three players were out with injuries. Coach Antonio Conte also decided to rest striker Marco Vucinic.
Copenhagen has yet to win at home this season and the 10-time Danish champion is 10th in the 12-team top flight.
Last month, the club fired coach Ariel Jacobs and re-signed former manager Stale Solbakken, hoping he could improve the team's fortunes.