The Danes took the lead after 15 minutes through Nicolai Jorgensen, but were thereafter indebted to an outstanding display from goalkeeper Johan Wiland for a share of the spoils.
Wiland was beaten 10 minutes into the second half by Fabio Quagliarella but, as the game went on, he pulled off a number of fine saves to ensure the opening Group B encounter ended level.
After the match, Solbakken was frank in his assessment of his side's performance.
"Our first half was next to perfect," the Copenhagen boss said to reporters.
"As the second half progressed we got tired, the distance between the players got bigger and we started to make the kind of mistakes you make as the result of being tired and missing control.
"However, we had a great goalkeeper and our heart was in it all the way."
Captain Lars Jacobsen also weighed in, admitting that Copenhagen had enjoyed their fair share of luck during the game.
"I think everyone could see that we were not exactly unlucky," the right-back said.
"Our first half was okay, while we got under heavy pressure in the second. However, we had a fantastic goalkeeper, and that is what you need in a match like this.
"We realise that the three away matches will be tougher than this one was, but this draw gives our confidence a tremendous boost."
Former Juventus defender Olof Mellberg, now of Copenhagen, was delighted to claim a point against his old club.
"We have got a point against one of the best teams in the world, and we should be content with our performance and with the result," he said.
Copenhagen next face a daunting trip to the Santiago Bernabeu on October 2 to take on Real Madrid, who opened with a 6-1 demolition of Galatasaray.