However, substitute Asensio grabbed Real's victor three minutes from the end. Their previously solid defense crumbled late last month in a 6-2 defeat to archrival Feyenoord and Erik ten Hag's team slumped to a 1-0 defeat on Saturday to Heracles Almelo.
Real went into the match brimming with confidence from a run of six wins and one draw in its last seven matches.
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.
"His adaptation has been very fast", Solari said.More news: Meghan Markle's half-sister challenges duchess to lie detector test
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Ajax, in the last 16 of the Champions League for the first time since 2006, had lost its last six matches against Madrid.
Real created little in the first half.
Having had a Nicolas Tagliafico goal disallowed in the first half after a review, Ajax found themselves behind through Karim Benzema's 60th goal in the tournament.
Ajax claimed a foul in the build-up to the goal, but this time the VAR did not intervene and Real will take a slender advantage back to Madrid.
"I believe it's possible to win four", Courtois said at a press conference at the Johan Cruyff Arena.
But just before the hour mark Benzema made the breakthrough with his 60th Champions League goal on his 450th Real appearance.
Real Madrid coach Santiago Solari will have to decide between youth and experience as his side looks to take another step towards a historic fourth consecutive Champions League title on Wednesday.