APOEL held out until the 74th minute when Kaka, on as a second-half substitute, crossed for Benzema to score the opening goal.
Kaka tapped in the second in the 82nd after Marcelo went on a determined run to the byline and managed to cut the ball back as he was falling.
Benzema, who missed an open goal in the first half, added his second goal in the last minute of normal time when he connected with a perfect pass from Mesut Oezil.
Madrid controlled play from the start, besieging the APOEL goal and forcing the Cypriot team into a defensive shell to stifle Madrid’s potent attack, which has scored an average three goals in each of their eight previous Champions League matches.
A volley from an unmarked Oezil from inside the box in the 12th was pushed away by APOEL goalkeeper Dionisios Chiotis.
Cristiano Ronaldo’s header from inside the box three minutes later sailed just over the crossbar. In the 28th, Ronaldo latched onto a Fabio Coentrao through pass inside the box, but his high, angled shot was well saved by Chiotis.
Benzema looked certain to give the visitors the lead their dominance deserved in the 34th when Nuri Sahin delivered a pass across goal – only for the French striker to scoop the ball over the bar from close range and with the goal gaping.
Ronaldo missed again in the 43rd when his flick from Oezil’s pass rolled wide.
Nine-time champion Madrid continued to be frustrated in the second half.
Ronaldo misses with a long-range effort in the 49th before his free kick in the 67th cannoned off the wall and fell for Oezil, whose shot was blocked.
APOEL, meanwhile, never threatened the Madrid goal and were limited to a few attacking forays by Brazilian forward Ailton Almeida.
Madrid finished the match with 20 attempts on goal, to APOEL’s zero, but it took the introduction of Kaka in the 64th minute to provide the spark for Madrid to break the deadlock, the Brazilian setting up the first goal and scoring the second before Benzema wrapped up a commanding away win.
APOEL, the first Cypriot team to reach the Champions League quarterfinals, suffered its heaviest loss so far in its historic campaign.