Asensio netted in the 87th minute from Dani Carvajal’s excellent cross to give Real the advantage heading into the return leg on March 5.
Karim Benzema swept holders Real into the lead in the second half, before Hakim Ziyech scored a deserved equaliser for an energetic Ajax, who created more than enough chances to win comfortably on the night.
They were also denied a first-half opener when Nicolas Tagliafico headed home only for the goal to be chalked off after a VAR check in a Champions League first as Dusan Tadic was judged offside.
Real goalkeeper Thibaut Courtois, whose blushes were spared by the VAR decision after his error led to the disallowed goal, felt the officials got it right.
"The correct decision? I think so," he told reporters. "When it happened I thought it could be offside. When the guy headed it I wanted to go and catch it.
"Because of his (Tadic's) intention I made a quick movement. Luckily for VAR because nobody would have seen it."
DAUNTING TASK
Sergio Ramos, making a 600th appearance for Real, backed the intervention of VAR and warned Ajax they will face a daunting task in the second leg.
"I have said many times that I am a great defender of the VAR because it will make football fairer little by little," Ramos said.
"We knew how to suffer, read the game and create spaces against them to take advantage of our speed. We achieved a great result and now it's up to them to suffer in our stadium."
The home side thought they had profited from a horror error by Courtois, who allowed a Donny van de Beek header to spill from his grasp and Tagliafico nodded the rebound into an empty net.
Yet referee Damir Skomina called for a VAR check and after a number of replays, ruled that Tadic had been standing in an offside position when the goal was scored.
Benzema put Real in front when he netted on the hour-mark, but Ajax got the goal their play deserved with 15 minutes remaining as Ziyech side-footed home after a sweeping move.
Yet Asensio had the final word, scoring at the back post from Carvajal’s cross.
"As a team we did well, it was too bad that our goal is rejected. Perhaps something like that is easy to do in favour of the big club," Ajax midfielder Frenkie de Jong said.
"But we have continued well and have received our opportunities afterwards. Too bad it did not yield a better result."
(Reporting By Nick Said; Editing by Toby Davis)
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