The nine-times European champions dominated the match but APOEL, the first Cypriot side to reach the last 16 of Europe's premier club competition, defended solidly to frustrate their Spanish opponents until the last 15 minutes.
The arrival of Kaka as a substitute midway through the second half created the breakthrough, the Brazilian producing a superb cross from the left for the diving Benzema to head powerfully home after 74 minutes.
Kaka slotted in the second goal from eight metres following a surging run down the left by his fellow substitute Marcelo and Benzema struck again in the last minute to put the La Liga leaders firmly on course for the semi-finals.
"It took us a long time to open the score but that had a lot to do with our opponents who played very well," Real coach Jose Mourinho told reporters.
"They don't have the outstanding individual talent that other teams in the Champions League have but in terms of organisation and character they are fantastic," the Portuguese added.
"They created many difficulties and I think we won the game because we played well. What they did was fantastic, and we had to play at optimum strength to win, they were fantastic on defence and they deserve our respect."
APOEL's Brazilian defender Marcelo Oliveira was carried off on a stretcher after pulling his hamstring early in the match but the home side initially made light of his absence.
Benzema missed a golden opportunity to open the scoring late in the first half, the French striker inexplicably directing the ball over the bar from close range with the goal gaping.
APOEL failed to register a shot on the Real goal but the Spanish side became increasingly frustrated by their inability to break through.
Cristiano Ronaldo caused problems without going close to scoring and it needed the introduction of Kaka to provide the spark for Real.
The playmaker cleverly set up the first goal for Benzema with a pinpoint cross and clinically converted Marcelo's cutback before Mesut Ozil crossed for Benzema to tap in his second.
APOEL coach Ivan Jovanovic said his side had become frustrated after the first goal.
"The truth is we were well organised, at least on defence, but certainly, some disappointment set in after the 75th minute. There was some disappointment and I would say afte the goal we lost some concentration," he told Sigma Sports.
APOEL chairman Phivos Erotocritou said APOEL were proud of their achievements.
"We have our head held high. It has been really special, we are very proud that we got to this point. It is certainly not a result to be ashamed of, we are very very proud we got to this point."
The second leg in Madrid takes place next Wednesday.
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