Los Blancos refused to be beaten, however, and Asensio levelled the scores with a free kick in the 83rd minute.
Real Madrid were officially presented with last season's La Liga trophy before the match at the Bernabeu, with Cristiano Ronaldo among those involved in the celebrations, but suspension ruled the Portuguese out once again.
A lively start to the match saw both teams move the ball at pace, but it was Real Madrid that made the breakthrough in the 10th minute when Asensio danced to the edge of the Valencia box before finding the bottom corner with a super finish.
The response from Valencia was impressive, however, and the visitors levelled the scores in the 18th minute when Toni Lato broke down the left before delivering an inch-perfect cross into Soler, who could not miss from close range.
Benzema might have sent Real Madrid back ahead in the 21st minute after a slick passing move involving Asensio, but the Frenchman hit the side-netting during an end-to-end period of the match.
Marcelo was next to come close for Los Blancos in the 33rd minute, with the Brazilian, who had been released by Asensio inside the Valencia box, just missing the crossbar with a powerful effort.
Benzema again came close in the 36th minute after the Frenchman had dribbled into the Valencia box following a counter-attack, but the striker could not find a route past Neto in the Valencia net.
Benzema had two more opportunities in the latter stages of the first half, but he volleyed over the crossbar in the 43rd minute, before heading wide less than 60 seconds later as the two teams headed down the tunnel on level terms.
Mateo Kovacic replaced Isco at the break as Zinedine Zidane looked to gain greater control of the midfield, and the alteration appeared to work as the home side began the second period on the front foot.
Casemiro worked Neto with a strike from distance in the 50th minute, before Marcelo had a dangerous cross blocked by Ruben Vezo moments later.
The pressure was relentless as Benzema met another Marcelo cross in the 52nd minute, only to turn his effort wide of the post, before Luka Modric volleyed over the crossbar from close range.
Valencia were a threat down the other end, however, with Kondogbia, Soler, and Zaza all causing problems in the final third.
Bale should have sent Real Madrid ahead in the 66th minute after meeting a super cross from Marcelo, but the Welshman's header somehow landed the wrong side of the post. Neto was then called into action to keep out a low strike from Modric as the match headed for its final 20 minutes at 1-1.
Navas could not prevent Los Che from taking the lead in the 77th minute, however, with debutant Kondogbia finding the bottom corner with a composed finish after the ball had broken for the former Sevilla midfielder in a dangerous area.
Asensio came up with an equaliser six minutes later, however, after wrong-footing Valencia goalkeeper Neto with a free kick.
Benzema had a golden chance to score a third for Real Madrid in the 86th minute after meeting a cross from Asensio, but the centre-forward fired over the crossbar, before missing an even better opportunity two minutes later - somehow turning wide from a central position.
Benzema was to have another brilliant chance to win it for Los Blancos, but the Frenchman saw his header tipped onto the post by Neto as Valencia rode their luck to secure a share of the spoils.