Yet Asensio had the final word, scoring at the back post from Carvajal's cross.
Real coach Santiago Solari, whose predecessors Zinedine Zidane and Julen Lopetegui both used young players only when they had to, has made youth his first choice since being tasked with rejuvenating his team in November. "He did everything he had to do from the moment he came into the pitch".
"Frenkie is fit", Ajax coach Erik ten Hag said earlier on Tuesday.
"Amazing atmosphere. I congratulate the fans and this team, who play very good football and with a lot of personality", he said.
The visitors came out firing in the second half and they almost went ahead after 50 minutes when Dani Carvajal played in Karim Benzema, who saw his first-time strike saved by Andre Onana. We did, but you have to be more ruthless. We had to stick together and remain united and make the most of the counter-attacks to break their line of pressure and we scored two goals, which is important in terms of the tie.
However, his goal was swiftly chalked off. VAR ruled out the goal with Dusan Tadic being deemed offside. "I think it's brilliant how he carries out his job as leader, which isn't an easy one".
"His adaptation has been very fast", Solari said.
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"I hope there will be many more and every day I get up excited to represent the team as captain".
In Amsterdam, Benzema struck a powerful shot past Andre Onana on the hour after a strong run from the left by Vinicius Junior to put the 13-time European champion ahead.
There is also a view that Bale can be very effective using his pace against exhausted rivals in the last 20 minutes of games as he did on Saturday. Ziyech's shot missed power and direction and hit Real goalkeeper Thibaut Courtois.
Real created little in the first half.
Vinicius chased a ball over the top down the left wing and cut inside before finding Benzema, who sent an unerring finish across Anona and high into the net.
Real Madrid advanced to the Round of 16 after finishing first in Group G, while Ajax advanced to the Round of 16 after finishing second in Group E.
Ajax might have felt optimistic when they drew an out-of-sorts Madrid back in December but Santiago Solari's side have found their rhythm since, winning six out of their last seven matches.