The result moved Zinedine Zidane's side above Barcelona into second, two points behind leaders Atletico Madrid with just two games remaining in the season.
Luka Modric got the club on its way early on with his fourth league goal of the campaign – the first time he has achieved that since 2011-2012 with Tottenham in the Premier League – before Rodrygo doubled its advantage on the stroke of half-time.
Jorge Molina threatened to set up a dramatic finale with a goal 19 minutes from time, but Los Blancos comfortably sealed an 18th win in their last 19 games against Granada thanks to goals from Alvaro Odriozola and Karim Benzema.
Madrid started on the front foot and almost went ahead after 14 minutes, Benzema's header forcing a smart stop from Rui Silva.
Zidane's men opened the scoring three minutes later, though, when Modric latched onto Miguel Gutierrez's sumptuous flicked ball over the top and rolled through Silva's legs from a tight angle.
The visitor's dominance was rewarded again in first-half stoppage time when Rodrygo powered into the penalty area down the right-hand side and fired across Silva for his first LaLiga goal of a frustrating campaign.
Granada reduced the deficit inside the final 20 minutes, Molina stroking into an empty net after Thibaut Courtois had parried Luis Suarez's strike into his path.
Substitute Odriozola settled any Madrid nerves, though, powering home after Eden Hazard's cross had fallen kindly to him in the 75th minute.
Benzema added gloss to the scoreline a minute later, expertly into an unguarded net from 35 yards after Silva's slapstick attempt to cut out Casemiro's long ball over the top.