Valencia debutant Geoffrey Kondogbia appeared to have snatched all three points for the visitors, but Asensio assured Madrid were not left empty-handed with his second goal of the match earning a share of the spoils.
The young attacker's first came after just 10 minutes, giving Madrid further reason to celebrate after it was presented with the LaLiga trophy after its triumphant 2016-2017 campaign prior to kick-off.
Asensio came into the side due to Sergio Ramos' suspension - Cristiano Ronaldo was also banned - and the defender's absence was evident when Carlos Soler darted between the makeshift centre-back pairing of Casemiro and Nacho to equalise in the 18th minute.
Karim Benzema wasted numerous chances to restore Madrid's lead and his profligacy was punished when Kondogbia guided home a cool finish with 13 minutes remaining.
However, Asensio picked out the bottom-left corner with an excellent set-piece from 20 yards five minutes later to bring Madrid level before Benzema had a stoppage-time header pushed onto the post as Valencia goalkeeper Neto ensured they lost ground to title rival Barcelona.