Riyad Mahrez's goal five minutes before the break proved enough to earn the Premier League champions a third successive win in Group G and they now only require one more win to guarantee their place in the last 16.
Leicester went into the game having suffered a one-sided 3-0 defeat at Chelsea at the weekend, and they were second best during the opening stages as Copenhagen dominated possession.
Copenhagen were looking a bigger threat in the final third but they struggled to create any clear-cut opportunities and an uneventful first half came to life with five minutes left as Leicester went ahead.
A cross from Vardy found Islam Slimani at the back post and after heading the ball back towards Mahrez, the playmaker was able to flick the ball into the net from six yards.
Copenhagen almost found an immediate response as Youssef Toutouh saw an effort deflected just over the crossbar while from the resulting corner, Andreas Cornelius flashed a header marginally wide of the far post.
Leicester returned for the second half on the front foot and Mahrez should have done better with a first-time strike from the edge of the penalty area within 60 seconds of the restart.
Back up the other end, Cornelius's flick went too close to Schmeichel before Mahrez saw another shot blocked by the defender after he had glided past two defenders on the edge of the box.
Leicester should have had a second goal with 23 minutes remaining but after Slimani had found the back of the net after his initial shot had been saved, he was wrongly adjudged to have been offside.
Despite that let-off, Copenhagen were still unable to mount any kind of sustained attack on the Leicester goal but with a minute remaining of normal time, Kasper Schmeichel pulled off an outstanding one-handed save to prevent Cornelius's shot from eight yards finding the bottom corner.
It was a match-winning intervention from a player who was born in the Danish capital and his side now go into the second half of the group stages needing just one more victory to continue their fairytale run.