A brilliant start saw Cristiano Ronaldo give the European champions the lead, slotting home after just two minutes, and Rodriguez added a second with a vicious volley 15 minutes before the break.
Karim Benzema added a third shortly after half-time, coolly poking his effort home following Ronaldo's flick.
Marcelo was denied by the crossbar four minutes later, before Rodriguez took advantage of a defensive mix-up to poke home a fourth and double his tally as Carlo Ancelotti's side followed up last weekend's Clasico triumph in style.
The result represents Real's 11th consecutive win in all competitions, while condemning a toothless Granada to their fifth defeat in six games.
The visitors, who named an unchanged side from the the convincing win over Barcelona, did not take long to assert their authority in Andalucia, with Ronaldo opening the scoring early on.
Some good work down the right saw Daniel Carvajal dispossess Dimitri Foulquier, which allowed Benzema to cut back to the Portugal international inside the area and he marked the 700th match of his career with his 22nd goal of the season.
Granada came into the game having mustered just one win in their last eight matches and posed little threat to Ancelotti's men, who continued to press.
A wonderful clipped ball from Isco allowed Benzema to force a smart save from Roberto in the 23rd minute, before Carvajal blazed his shot over the bar from six yards following Ronaldo's brilliant backheeled pass.
But the lead was doubled just after the half hour, when a superb sweeping move saw Benzema tee the ball up for Rodriguez inside the area and the Colombian smashed an unstoppable first-time volley into the top corner.
Real's dominance continued in the second half, and they were celebrating a third just nine minutes after the restart, Ronaldo finding Benzema with another audacious backheel, allowing the Frenchman to beat the offside trap and prod his effort past Roberto.
Granada's goalkeeper was once again a spectator two minutes later and had the crossbar to thank as Marcelo's 25-yard effort was denied.
The home side continued to struggle, but went close to pulling a goal back after 65 minutes.
Substitute Ruben Rochina fired over with an acrobatic volley, before a whipped cross from Juan Carlos was headed over by Abdoul Sissoko.
Fellow sub Javier Marquez should have done better six minutes from time, but was denied by an onrushing Iker Casillas.
Rodriguez wrapped up the win three minutes later by turning home at the far post following some neat play from Ronaldo, capping off a routine afternoon for Ancelotti's charges.
