Madrid had 21 shots at Estadio Alfredo di Stefano but had been frustrated prior to the breakthrough strike from Asensio four minutes before half-time, with Eibar goalkeeper Marko Dmitrovic in inspired form.
Dmitrovic finished with six saves, while three of Madrid's four offsides saw goals ruled out, yet Karim Benzema secured the points with a second-half header that extended his scoring streak to seven straight games.
Benzema has plundered nine goals in that time, while he has nine against Eibar in his LaLiga career - the most of any Madrid player.
Asensio is enjoying his own fine run, netting in three consecutive matches for Madrid for the first time, and now has his sights set on more important fixtures to come.
Liverpool visit in the Champions League on Tuesday before Madrid welcome rivals Barca next Saturday.
"I work for that [to score]," Asensio told Movistar. "In the end, it's about waiting for the moment and taking advantage of it.
"I'm happy for the goals, for the team and now comes the most important moment of the season.
"There are unfortunately a lot of injuries this season, but I think we all have to contribute and that's why we're a team."
Victory took Madrid up to second, at least until Barca play Real Valladolid on Monday.
Asensio said: "We knew we had to win to get good feelings and we took the three points, which is the most important thing.
"We have played a good match, creating chances, and in the end we have won well.
"Now we face a very important week with the match against Liverpool and then Barca to come. We have to live up to it as the Madrid we are."