Ajax pounced early on a defensive lapse in the box when Franklin Cale crossed the ball into the path of Tashreeq Morris who tapped in for an easy goal in the ninth minute.
After conceding early, Celtic fought back almost immediately when blistering winger Thapelo Morena found some space in an advanced position to unleash a shot with the side of the foot, but his attempt flew over the target.
Ajax showed no intention of defending the slender lead and were out for more goals in what would turn into an open and exciting encounter.
Toriq Losper brilliantly beat his marker to accelerate up the pitch with the ball and played through Erwin Isaacs inside the box.
Isaacs took two deft touches to make space for a shot that flew straight into the arms of Celtic goalkeeper Patrick Tignyemb in the 18th minute.
It was a battle at the back between Ajax defender Rivaldo Coetzee and Celtic striker Lerato Lamola as they tussled for the ball near the corner flag.
The tricky Lamola got the better of his opponent to whip in a cross into the box, but there was no teammate wiling to complete the move.
Ajax were soon in with another scoring opportunity when Bantu Mzwakali combined with Cale and powered through the opposition wall to shoot and force Tignyemb into a superb save in the 23rd minute.
In the following exchanges both teams threatened each other with attacks but none was able to find the back of the net.
At the end of the half it was advantage Ajax, but the visitors were beginning to mount a fight-back in hunt for the equaliser.
In the second half Ajax showed that they could also defend as they contained the opposition attack after some sustained pressure.
In the 73rd minute Celtic midfielder Musa Nyatama was the instigator in a move that saw the ball brush the crossbar after an effort from Lamola.
Playing at home in front of an expectant crowd, Ajax held onto their lead for dear life and marched into the last eight.
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