Los Blancos struggled to find a way into the game early on against an Elche side that was set up in a very defensive fashion.
The home side were left even more frustrated after Fran Garcia and Aurelien Tchouameni both had penalty appeals turned down after stiff challenges in the box.
They found an opening in the 39th when a Federico Valverde free kick wasn't cleared properly and Antonio Rudiger volleyed the loose ball past Matías Dituro, putting the home side ahead.
Around five minutes later, the winger continued his fine form by getting on the scoresheet himself with a lovely curling shot into the top corner.
Dean Huijsen put the result beyond doubt in the 66th minute when he got on the end of a Daniel Yanez cross to head Real Madrid into a 3-0 lead.
Elche managed to pull one back in the 85th minute via a Manuel Angel Moran own goal before Arda Guler restored Los Blancos' three-goal advantage with one of the goals of the season.
Guler had possession in his own half when he saw Dituro off his line and hit a fine long-range shot from downtown into the back of the net.
Elsewhere in LaLiga on Saturday, David Costas' 30th-minute strike was enough to hand Real Oviedo a 1-0 win over Valencia at Carlos Tartiere Stadium.
Atletico Madrid got ahead against Getafe early, with Nahuel Molina scoring in the eight minute, and clung on for a 1-0 win at Riyadh Air Metropolitano.
Girona showed their class in a comfortable 3-0 win over Athletic Club at Estadi Montilivi.
Hugo Rincon opened the scoring in the fourth minute, with Azzedine Ounahi adding a second in the 77th minute. Claudi Echeverri rounded out the scoring with a strike in stoppage time.
Nicolas Pepe rescued a point for Villarreal against Alves at Estadion de Mendizorroza with a strike deep into stoppage time after a Rafa Marin own goal in the 40th minute put the home side ahead.
Real Madrid have received a significant lift ahead of their Champions League round-of-16 second-leg clash with Manchester City on Tuesday, with both Kylian Mbappe and Jude Bellingham included in the squad traveling to England.
Atletico Madrid manager Diego Simeone has launched a strong defence of his team and striker Julian Alvarez, criticising the media for focusing on transfer speculation rather than Atletico's performances.
Real Madrid hat trick hero Federico Valverde praised his team's unity in the 3-0 UEFA Champions League Round of 16 first leg victory against Manchester City on Wednesday.
Manchester City face a huge challenge to keep their Champions League hopes alive after Federico Valverde's first-half hat-trick gave Real Madrid control of their last-16 tie.
