The forward forced an own goal and scored the second as Madrid moved into third with 20 points, one behind second-placed Valencia and one ahead of Atletico Madrid, who beat Celta Vigo 1-0 earlier on.
Asensio, picked ahead of Karim Benzema to partner a subdued Cristiano Ronaldo up front, sparkled as Real made light work of the visiting Basque side, who are 16th.
Eibar's Paulo Oliveira headed Asensio's cross into his own goal in the 18th minute before the forward volleyed home the second goal 10 minutes later from Isco's ball.
Substitute Marcelo added the third in the 82nd with a low drilled effort into the far corner to wrap up the victory.
(Reporting by Rik Sharma; Editing by Ken Ferris)
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