Substitute Kaka sparkled and Karim Benzema scored twice as Real Madrid finally broke down the Cypriot underdogs Apoel in Nicosia.
The Brazil international was introduced in the 63rd minute and set up Benzema for a 74th-minute opener before tucking away fellow sub Marcelo's excellent cutback eight minutes later.
Benzema's second gave the scoreline an emphatic look despite a mighty defensive effort by Apoel, whose giant-killing run has accounted for Lyons, Porto and Shakhtar Donetsk but who were outclassed last night.
"It took us a long time to open the score but that had a lot to do with our opponents who played very well," the Real coach, Jose Mourinho, said. "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.
"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 lost key defender Marcelo Oliveira to injury inside 10 minutes after his standing leg twisted under him as he overcommitted in a tackle. Mesut Ozil tested goalkeeper Dionisis Chiotis, with the hosts still down to 10 men before replacement centre-half Kaka – the namesake and compatriot of Real's in-form schemer – could be introduced.
Fabio Coentrao shot well wide and Benzema was guilty of a shocking miss in the 33rd minute when he hung a foot out at Cristiano Ronaldo's low cross but stubbed the ball into the ground and it bounced over the bar.
Mourinho was driven to a double change in an attempt to lift his side, introducing his own Kaka and Marcelo for Gonzalo Higuain – injured a few minutes previously – and Coentrao.
Soon Kaka crossed deliciously from the left and Benzema dived to head past Chiotis. The keeper saved Ronaldo's drive with his legs and then fumbled Kaka's effort as Real sought a killer second goal.
It duly arrived as Marcelo was caught in the box by Paulo Jorge but kept his balance and cut the ball back for Kaka to slot home. Benzema tucked away Ozil's inviting low pass in the last minute as Real put the tie to bed ahead of the 4 April return leg.
Apoel Nicosia (4-4-2): Chiotis; Poursaitides, Oliveira (Kaka, 13), Jorge, Boaventura; Morais, Pinto (Solari, 72), Charalambidis, Trickovski; Alexandrou (Sousa, 46), Almeida. Substitutes not used Urko, Adorno, Solomou, Marcinho.
Real Madrid (4-2-3-1): Casillas; Arbeloa, Pepe, Ramos, Coentrao (Marcelo, 63); Khedira, Sahin (Granero, 84); Ozil, Higuain (Kaka, 63), Ronaldo; Benzema. Substitutes not used Adan, Altintop, Albiol, Varane.
Referee F Brych (Germany).