The Spanish giants, who are nine time winners of Europe's premier competition, dominated the possession stats in the first half but were frustrated by stubborn APOEL defending.
When Madrid were able to penetrate the hosts' defence, Benzema was guilty of missing a golden chance as he shot into the ground and over the bar from just two yards.
The Frenchman made amends when he eventually opened the scoring for Jose Mourinho's side with a 74th-minute strike.
Substitute Kaka then made Real's lead a comfortable one by applying the finish to a lovely Madrid move. The Brazilian was on hand to convert from six yards following smart build-up play from Marcelo and Cristiano Ronaldo.
Benzema put the memory of his miss firmly behind him with his second and Real's third in injury time. Mesut Ozil's pass found the striker in space at the back post and he struck home to complete the scoring.