The German club's position as Group G runners-up had already been secured ahead of the final matchday, and they were unable to sign off with a victory, as the hosts came from behind to secure a share of the spoils.
Copenhagen had a golden chance to take the lead in the ninth minute when Mohamed Daramy delivered a cross onto the head of Roony Bardghji, but his effort was saved by Gregor Kobel.
Dortmund's first opportunity arrived in the 17th minute through Donyell Malen, but the attacker could over fire over the crossbar after smart work from Giovanni Reyna.
BVB did make the breakthrough in the 23rd minute, though, with Thorgan Hazard collecting a cross from Felix Passlack before finding the back of the net with a smart finish.
Copenhagen were causing problems of their own, and Kobel had to make an important save to keep out Viktor Claesson's effort in the 29th minute, but the home side levelled just before the interval, with Hakon Arnar Haraldsson firing into the far corner after smart work from Claesson.
The away side had a golden chance to regain the lead in the 52nd minute when Karim Adeyemi found Malen inside the box, but the latter somehow missed the target.
Youssoufa Moukoko hit the post for Dortmund in the 64th minute, and BVB were looking the more likely to score a winner as the match headed towards its conclusion, but the spoils were ultimately shared, with the visitors finishing in second, while Copenhagen have ended their campaign in fourth behind Sevilla.