Neil Lennon's men led 1-0 at half-time through Odsonne Edouard but the home side hit back after the break, scoring through Dame N'Doye before having the chance to win it from the penalty spot but Jens Saget's effort was saved by Fraser Forster.
The sides reconvene at Parkhead on Thursday for the second-leg of the Eureopa League last-32 match-up with the tie in the balance.
Both sets of fans displayed pre-match tifo displays, while at one during half-time they appeared to be in competition to 'out-party' one another.
The travelling support were widely praised by the city's police.
The Kobenhavns Politi tweeted: "We want to thank the fans of @CelticFC - it's been great having you guys in town - no registered problems during the night. We wish you a safe journey back to Scotland."
Meanwhile, Copenhagen sent the club a tweet stating their excitement for the return leg.
It said: "Dear @CelticFC It was a pleasure to host you tonight. We look forward to visiting you for another exiting match. Have a safe trip home."