Asensio opened the scoring at the Santiago Bernabeu with a lovely finish, but Valencia hit back through Carlos Soler and took the lead when debutant Geoffrey Kondogbia slotted past Keylor Navas with 13 minutes remaining.
However, Asensio, elevated to the starting line-up due to Sergio Ramos' suspension, tucked home an excellent free-kick in the 82nd minute to earn the La Liga champions a share of the spoils.
"There is no doubt that Asensio is already the present of Madrid, not the future," Carvajal told Spanish television after the 2-2 draw.
1 - Marco Asensio has scored his first direct free-kick with his second attempt for Real Madrid (all competitions). Hero. pic.twitter.com/LrThx87pbL— OptaJose (@OptaJose) August 27, 2017
Karim Benzema wasted a host of chances and saw a stoppage-time header tipped onto the post by Valencia goalkeeper Neto, with Carvajal ruing their missed chances.
"We have not won one point, we have lost two," he said.
"We got a point and we have paid for the intense month that we have had. We had plenty of chances and we could have achieved something more.
"We let two points get away from us and we had clear chances in which the goalkeeper made a save or Benzema had bad luck."