Los Blancos are now on an 11-game win streak in all competitions. In the process, Real Madrid's title defence is back on track, as they trail Atletico Madrid by only three points, albeit having played a game more.
Meanwhile, Eibar sit 19th in the La Liga and are now 13 league games without a win. With only ten games remaining in the season, Jose Mendilibar's men find themselves thick in the relegation battle.
Real Madrid began the game on the front foot and quickly established their dominance. In the first five minutes of the game, Real Madrid had the ball in the back of the Eibar net, but the effort was ruled out for offside.
Nevertheless, Zinedine Zidane's men continued to pile pressure on their visitors, who struggled to get out of their own half. Real Madrid came close to scoring the opening goal, but Marco Asensio struck the frame of the goal from a free-kick.
After the half-hour mark, Real Madrid once again had the ball at the back of the net, this time by Asensio. But the goal was disallowed after VAR found the Spaniard in an offside position in the build-up to the goal.
Asensio was finally rewarded for his persistence in the closing stages of the first period as he gave Real Madrid a deserved half-time lead.
The second half began with Real Madrid threatening to double their lead. Eibar grew into the game and progressed the ball up the pitch. But poor finishing and lack of creativity meant that they could not break down a resolute Real Madrid defence.
Real Madrid had a goal disallowed for the third time in the game after Sergio Arribas was adjudged to be in an offside position in the build-up to Karim Benzema's goal. Madrid, however, doubled their lead two minutes later when Benzema scored with a header from inside the area.
Eibar turned up the tempo in the final minutes of the game but could not get a consolation goal as Real Madrid deservedly walked away with all three points.
The result sees Real Madrid pip rivals Barcelona to second place in the league table. On that note, let's have a look at the Real Madrid player ratings.
Thibaut Courtois - 7/10
It was the first league clean sheet for Real Madrid custodian Thibaut Courtois since January. He was fortunate not to concede a bizarre goal, as a brief lapse in concentration almost saw the ball go into the net. Courtois made four saves on the night.
Lucas Vasquez - 7/10
Lucas Vasquez found himself in acres of space down the right as Real Madrid focused the bulk of their attacks down the other flank. The Spaniard did well to exploit his space whenever he got the ball and was a vivacious figure all game.
Eder Militao - 7/10
It was an excellent performance from the Brazilian. Militao won a team-high seven duels and was also very tidy in possession, completing all but one of his attempted long balls.
Nacho Fernandez - 7/10
Nacho Fernandes was very diligent defensively, completing seven clearances, the most by a Real Madrid player on the night. His reading of the game was astute, as he was always on hand to thwart Eibar's attacks.
Ferland Mendy - 7/10
Ferland Mendy was played in a more central role than his usual wide position, as he featured on the left side of the back three. Mendy, however, showed superb composure both on and off the ball.
Marcelo - 7/10
The Real Madrid captain on the night put in an outstanding performance in just his seventh start of the league campaign. Marcelo was always an attacking outlet on the left before he was removed after 68 minutes.
Casemiro - 7.5/10
Casemiro provided the assist for Real Madrid's opener with a simple pass to Asensio. He was very energetic defensively and always looked to pounce on loose passes in midfield.
Luka Modric - 7/10
The Croatian had a quiet evening but was diligent regardless. Luka Modric kept play ticking for Real Madrid in his usual fashion. He completed a team-high ten long balls.
Isco - 7/10
Isco made just his sixth start for Real Madrid in the La Liga this season but put on a fine show. He was a creative force for his team, creating five chances, the most by a Real Madrid player on the night.
Marco Asensio - 7.5/10
Marco Asensio did it all in the first 45 minutes - hitting the post and having a goal disallowed for offside before finally scoring the opener for Real Madrid. The Spaniard received a pass from Casemiro and executed an exquisite touch which sent him through on goal before he bent one past the Eibar goalkeeper.
Karim Benzema - 7.5/10
Karim Benzema should have given Real Madrid the lead very early in the first half but squandered the chance. He then scored once in each half, but both efforts were ruled out for offside. Benzema finally got his goal with a thumping header from a cross from the left.
Ratings of Real Madrid substitutes
Toni Kroos - 7/10
Toni Kroos came on for Modric just after the hour mark. The veteran midfielder completed all his passes during his time on the pitch.
Rodrygo - 6.5/10
Rodrygo came on for the final 30 minutes alongside Kroos but could not do much, as his flank was scarcely used.
Sergio Arribas - 6.5/10
The 19-year-old Real Madrid academy graduate was very tidy in possession; Sergio Arribas also progressed the ball well.
Vinicius Jr - 7/10
Vinicius Jr. got the assist for the crucial second goal for Real Madrid. He received the ball from just after the halfway line and advanced upfield before pinging a lovely ball for Benzema to head home.
Mariano Diaz - 6.5/10
Mariano Diaz came on for Karim Benzema for the final ten minutes of the game.