Ronaldo, banned for pushing a referee after being sent off in the first leg of the Spanish Super Cup, opened the scoring after just 12 minutes from Gareth Bale's low cross.
The Portuguese doubled his tally from the penalty spot before Sergio Ramos's overhead kick rounded off the scoring.
Real move to the top of Group H alongside Tottenham Hotspur, who ended their Wembley hoodoo with an impressive 3-1 win over Borussia Dortmund.
Real boss Zinedine Zidane admitted that the club's failure to replace Alvaro Morata had left his striking options limited, particularly in the absence of Karim Benzema through a hamstring injury.
More From This Section
However, with Bale excelling back on the left, Ronaldo filled that void as he swept home at the back post from the Welshman's cross.
Ronaldo then headed into the side-netting from a similar position from another Bale delivery moments later.
And the hosts' rhythm was disrupted as they struggled to create more clear-cut chances against the dogged Cypriot champions before the break.
Ronaldo thought he had doubled his tally a minute into the second half as his volley from Dani Carvajal's pinpoint cross came off the underside of the bar, but didn't cross the line despite his protests.
Madrid's luck turned moments later when Roberto Lago was very harshly penalised for handball from Bale's deflected cross and Ronaldo smashed home the resulting penalty.
However, it was Ramos who struck next in acrobatic fashion from point-blank range.
Ronaldo thought he had bagged an eighth Champions League hat-trick when he swept home Luka Modric's cross, but was denied by the offside flag as Real had to settle for three as they began their quest for a 13th European Cup.