Granada put up a brave fight at the Bernabeu but went down 1-0 courtesy of a second-half strike from Karim Benzema.
Keeper Andres Fernandez was on hand to prevent Cristiano Ronaldo in the first half, and Granada should have taken the lead shortly after when Youssef El-Arabi found the back of the net, but he was harshly adjudged to have been offside.
Granada had a golden chance to go in front early in the second half through Isaac Success, but his shot was wide and they were soon made to pay when Benzema headed home a superb cross from Isco in the 55th minute.
Real had several chances to kill off the match late in the game, but despite missing out they managed to secure all three points on the night.
Valencia 0-0 Real Betis (Mestalla)
Real Betis were reduced to 10 men but still managed to frustate Valencia as they earned a 0-0 draw away.
Sofiane Feghouli had two decent chances for the home side in the first half but couldn't convert.
Just after half-time, Betis midfielder Dani Ceballos was sent off for a second bookable defence, and while Valencia went in chase of a winning goal, they could not manufacture an opening as the match remained scoreless until the final whistle.
Eibar 0-2 Atletico Madrid (Estadio Municipal de Ipurua)
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Atletico manager Diego Simeone rang the changes in the second half and it paid off as Fernando Torres and Angel Correa both came off the bench to score in a 2-0 victory over Eibar.
With the home side controlling too much of the possession for Simeone's liking, he brought on Torres and Correa and it soon paid dividends.
For the first goal, which came just after the hour mark, Torres squared the ball inside to Correa, who beat the defender to put the visitors in the lead.
Fifteen minutes later, Correa played provider for Torres, who lifted the ball over the oncoming goalkeeper to score Atletico's second.
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Eibar nearly pulled one back near the end, but Atletico held on for the win, which puts them in second on the La Liga table.
Real Sociedad 2-3 Espanyol (Estadio Anoeta)
Espanyol midfielder Hernan Perez became a hero overnight as he struck a 91st minute winner to lead his side to a 3-2 win over Real Sociedad on Saturday.
While Espanyol emerged victorious in the end, it was Sociedad who made their intentions clear early on as they came close to scoring on numerous occasions.
Sociedad's pressure paid off in the 20th minute when striker Imanol Aggirretxe had no problems slotting the ball into the back of the net from close range after being set up by Alberto de la Bella.
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After conceding the goal, Espanyol began to show signs of life as Victor Alvarez, Enzo Roco and Gerard Moreno all came close to finding the equaliser. However, Espanyol finally brought themselves level in the 42nd minute when Moreno converted a penalty, which was given after Sociedad goalkeeper Geronimo Rulli was sent off for a challenge on Marco Asensio.
Espanyol had to wait until the 71st minute to take a 2-1 lead courtesy of Roco's header into the top left corner. But, Espanyol's celebrations came to an end in the 86th minute when Sociedad forward Jonathas latched on to Gonzalo Castro's corner and fired it into the goal.
With the game set to end as a 2-2 draw, Perez came to the rescue for Espanyol as he headed Alvaro Gonzalez's corner into the bottom right corner to hand his side three valuable points.