Wiland produced several superb saves to keep his side in the game after Nicolai Jorgensen had given Copenhagen a surprise lead in the 14th minute.
Jorgensen, the Denmark international, latched on to a loose ball in the area and poked home.
Fabio Quagliarella ended Wiland's resistance nine minutes into the second half, levelling the scores with a left-footed shot from Federico Peluso's cross.
With his side visibly tiring as the second half wore on, Wiland continued to frustrate Juventus with his heroics between the posts as the Italians attacked in waves.
"We should have scored a lot of goals," Quagliarella said. "They scored early in the game but then Buffon did not have to make a save for the whole game, while their goalkeeper made a series of great saves."
Swedish international Wiland was helped in no small part by wasteful finishing, with Arturo Vidal and Carlos Tevez both missing late chances that could have given Juventus victory.