The Foxes could have sealed progression into the last 16 with a win, but a goalless draw in Denmark leaves them with two games to complete the job.
The reigning Premier League champions currently sit at the top of Group G on 10 points, three ahead of second-placed Porto and five above Copenhagen.
The club are unbeaten in the competition so far and a fourth consecutive clean sheet guarantees them at least a third-place finish, which means that if they fail to reach the last 16, they will drop into the Europa League.
"It's a good point against a good team away from home, so I'm satisfied," Leicester's Twitter page quotes Ranieri as saying. "[It was] better from us in the second half, we just couldn't make the last pass. There were some fantastic duels in the air.
"Ten points is good, but not enough. Porto are close behind us. We have our destiny in our hands."
Leicester's next group match will be against Club Brugge on November 22.