Nicolai Jorgensen gave the hosts the lead in the 15th minute in the Group B clash before Fabio Quagliarella levelled the scores 10 minutes into the second half.
It was scant reward for a Juventus side that enjoyed much more of the ball and created far better chances, only for wasteful finishing to deny Antonio Conte's side.
Stale Solbakken made one change from the side that drew 1-1 with Esbjerg at the weekend, Igor Vetokele making way for Daniel Braaten.
Conte made three changes from Saturday's 1-1 draw with Inter, with Angelo Ogbonna, Federico Peluso and Quagliarella coming in for Andrea Barzagli, Kwadwo Asamoah and Mirko Vucinic.
The Italian champions had the first effort of note in the fifth minute, Carlos Tevez meeting Andrea Pirlo's cleverly-taken corner but firing wide of the right-hand post from an angle.
Copenhagen took the lead after Juventus failed to deal with Pierre Bengtsson's deep free-kick from the right-hand side.
As a crowd of players jumped to challenge for the ball inside the six-yard box, it fell kindly for Jorgensen, who took a touch and slotted home beyond a helpless Buffon.
Juventus' response was immediate as Johan Wiland made two outstanding saves in a minute to keep his side in front midway through the half, keeping out efforts from Giorgio Chiellini and Paul Pogba when both players looked certain to score.
Tevez then curled his shot a yard over the crossbar in the 32nd minute as Juventus continued to pen their opponents back in search of an equaliser.
The second half began much as the first had ended, with the visitors in the ascendancy, and in the 52nd minute Tevez wasted a golden opportunity to level the scores.
Pirlo's first-time ball over the top caught the home defence square, but the Argentine forward, clean through on goal, saw his effort saved by the legs of Wiland.
Two minutes later, however, Juventus finally got the goal their dominance had warranted as Tevez stepped over Peluso's cross from the left and Quagliarella swept the ball high past Wiland from 10 yards.
Quagliarella was then inches away from a second in the 65th minute, his header from Lichsteiner's cross flicking the top of the bar and bouncing away to safety.
With 20 minutes to go, Jorgensen went close in a rare Copenhagen attack, driving forward before unleashing a shot from 20 yards that flew narrowly over Buffon's crossbar.
Arturo Vidal could have put Juventus in front with 15 minutes remaining after collecting Tevez's headed flick, but his weak effort rebounded away off Wiland's chest.
Moments later, Tevez was again denied by Wiland before Pogba's header bounced just wide of the post with the goalkeeper stranded.
With 10 minutes to go, substitute Sebastian Giovinco was the next man to be denied by the inspired Wiland as Juventus were left searching in vain for a winner.
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