Victory cuts third-placed Real's deficit on Barcelona at the top of La Liga back to eight points.
However, it was another off night for Madrid's strikers as Cristiano Ronaldo and Karim Benzema were held scoreless as Casemiro, Isco and 21-year-old Asensio provided the killer touch.
Benzema has been fiercely criticised for his role as Real lost to promoted Girona last weekend and at Tottenham Hotspur in the Champions League on Wednesday.
Both Ronaldo and Benzema have only one La Liga goal each to their name so far this season and it was the World Player of the Year who had Real's next three efforts as he blasted into the side-netting, curled a shot just wide and then placed a header from close range too close to Lizoain.
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Madrid had to wait until four minutes before half-time to break the deadlock when Casemiro grabbed Real's first headed goal of the season from a corner.
The visitors had a penalty shout early in the second- half, but on-loan Atletico Madrid winger Vitolo went to ground too easily despite a slight pull of his shirt by Casemiro.
A booking would have ruled Casemiro out of the Madrid derby against Atletico in two weeks' time and he was quickly replaced after Asensio secured the three points in stunning fashion.
Asensio has made a habit of netting in spectacular fashion in his first 18 months at Madrid and pounced on a loose ball 30 yards from goal to rifle a shot that rose all the way on its trajectory into the top corner.
Elsewhere, Villarreal's unbeaten league run since Javier Calleja took charge in September continued as Nicola Sansone came off the bench to grab both goals in a 2-0 win over Malaga to move the Yellow Submarine up to fifth.
Real Sociedad were also comfortable 3-1 winners over Eibar. However, their Basque rivals Athletic Bilbao's slump show no signs of abating as they conceded three times in the opening 26 minutes to lose 3-1 at Celta Vigo with Iago Aspas netting twice for the hosts.
And Girona moved into the top half of the table by backing up their stunning upset of Real Madrid last weekend by edging out Levante 2-1 in the lunchtime kick-off.
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