Jose Mourinho's side badly needed a good performance after taking only a point from their opening two games and Ronaldo stepped up with his 149th and 150th goals for the club on his 149th appearance in all competitions.
The victory left Real five points adrift of arch- rivals Barcelona, who won 1-0 at home to Valencia thanks to a screamer from full-back Adriano and are the only team to have a perfect record after three games.
Mourinho voiced rare public criticism of his Real players after last weekend's shock 2-1 defeat at Getafe and he was still not satisfied after Sunday's success.
The Portuguese said he had been hoping to see the same intensity they showed in the opening 20 minutes of Wednesday's Spanish Super Cup against Barca, when they scored twice and held on for a 2-1 success.
"Today the team had little ambition, little rhythm, little intensity and I did not like it," Mourinho told a news conference.
"It's not enough just to get the points, the team has to play more and work more," he added.
Ronaldo's opening goal came in the 26th minute when his deflected left-foot strike from a narrow angle arrowed between the legs of Granada goalkeeper Tono.
Tono saved Ronaldo's first-time shot nine minutes into the second half and he tapped the rebound into an empty net before limping off after he was felled by Borja Gomez.
Ronaldo left reporters scratching their heads after the match when he said he had not celebrated his goals because he was feeling "sad" for "professional" reasons.
He refused to elaborate and added that he should have recovered from Sunday's knock in time for Portugal's World Cup 2014 qualifiers against Luxembourg and Azerbaijan next week.
Barca saw off Valencia with a dominant but uninspired display at the Nou Camp distinguished by Adriano's brilliant 23rd-minute strike.
"It was important to win this game as well before the international break," midfielder Andres Iniesta, who replaced Fabregas midway through the second half, told Spanish television.
"We probably should have scored more goals given the chances we had but we're happy with the win and this is the path we have to follow," added the Spain international, who presented the Best Player in Europe award he won on Thursday to fans before the game.
