The visitors tested goalkeeper Johan Wiland, whose saves include diverting a low shot by Ronaldo with his fingertips. Toward the end of the first half, a header by Copenhagen's Thomas Delaney finished in the Madrid net but the goal was disallowed by German referee Felix Brych due to a collision with keeper Iker Casillas.
Copenhagen's Rurik Gislason tried his luck with a shot in the early stages of the second half but the ball ended in the side-netting. Later, Casillas punched away another shot by the Icelandic midfielder.
However, Madrid was in complete control of the game, led by Ronaldo and backed by a solid defense. The home team, backed by a crowd of 37,000, tried in vain to force itself back into the match.
Ronaldo hit the post in the 75th minute, while teammate Gareth Bale missed another opportunity for Madrid's third goal in the 82nd minute as he surged into the box.
Seconds before the final whistle, Wiland punched away Ronaldo's powerfully-struck penalty kick.
While Madrid will play in the last 16, the Danish club finished bottom of the group and exits European competition for the season.
"It was a serious game that wasn't at all easy," Madrid coach Carlo Ancelotti said. "My players did well, and showed good attitude and solid play."