Marc Cardona actually sent Eibar ahead in the 39th minute, but Benzema netted twice in the second period to continue his fine form in front of goal, sending Madrid to within four points of second-placed Atletico Madrid in the process.
Madrid boss Zinedine Zidane made five changes to the team that started against Valencia last time out, with Nacho, Sergio Reguilon, Isco, Federico Valverde and Gareth Bale coming into the side. Sergio Ramos was left out altogether, while the likes of Casemiro, Toni Kroos and Lucas Vazquez dropped to the bench
The home side had a golden chance to take the lead in the sixth minute when Bale raced into a one-on-one with Eibar goalkeeper Marko Dmitrovic, but the Welshman saw his first attempt saved before hooking the rebound wide.
Benzema did have the ball in the back of the Eibar net in the ninth minute, but the offside flag came to the away side's rescue as the score stayed level.
Eibar's first serious chance of the afternoon arrived in the 10th minute when Marc Cucurella found space in a wide area before delivering a brilliant low cross towards Sergi Enrich, but the striker just could not make contact.
Both sides were having their moments as the first period reached its mid-point, but clear opportunities were few and far between with some misplaced passes in the final third of the field.
Bale struck harmlessly wide of the Eibar post in the 29th minute, but it was the visitors that incredibly took the lead in the 39th minute when Cardona converted.
Madrid's first shot of the second period came from Luka Modric in the 48th minute, but it was a comfortable save for Dmitrovic between the away side's sticks.
Down the other end, Cardona almost broke into the Madrid box once again, but Alvaro Odriozola just about did enough to make the clearance as the visitors - who also saw a header from Enrich miss the post - threatened a second.
Los Blancos thought that they had levelled in the 55th minute when Benzema converted following a mix up at the back, but once again the offside flag came to the away side's rescue at the Bernabeu.
Finally the home side equalised in the 59th minute, though, and the goal came from Benzema, who headed a brilliant Asensio cross into the back of the net.
Asensio was then denied by the Eibar goalkeeper as the home side looked to complete the turnaround.
The score was still level approaching the final 15 minutes, though, with Eibar recovering from the setback to form a strong defensive rearguard in the Spanish capital.
Lucas and Kroos both entered for the final 10 minutes or so, and Madrid scored a second - securing all three points in the process - when Benzema's header from a Kroos cross ended up in the back of the net.
The Frenchman had two chances to complete his hat-trick late on, but Dmitrovic was on hand to make a smart save before the forward somehow missed an open goal as it finished 2-1 to the European champions.
REAL MADRID (4-3-3): Navas; Odriozola, Varane, Nacho, Reguilon; Isco (Diaz 90'), Valverde, Modric (Kroos 77'); Asensio, Benzema, Bale (Lucas 77')
EIBAR (4-1-4-1): Dmitrovic; De Blasis, Ramis (Orellana 54'), Oliveira, Cote; Escalante; Cardona (Pena 73'), Alvarez, Jordan, Cucurella (Charles 85'); Enrich