They almost doubled the lead in the closing stages of the match when Stefano Denswil was left unmarked inside the box, but his header bounced onto the outside of the post.
Here, Sports Mole takes an in-depth look at an important night of Champions League football in Amsterdam.
On target 7
On target 2
Was the result fair?
Yes. As the stats show, Ajax enjoyed the lion's share of possession this evening and, while they struggled to use it productively at times, they certainly posed more of a threat than Celtic. The second half was more lively than the first with both sides looking capable of scoring, but it was Ajax who created the best chances throughout. In addition to the goal, they hit the post and had one cleared off the line, while Celtic's best moment was a tame Anthony Stokes effort shortly after they fell behind.
Given their recent form, their perilous Champions League position and the fact that they were beaten by Celtic just two weeks ago, manager Frank de Boer will be very happy with his side's performance. It is clear that they are not one of Europe's elite, but the footballing ideals of Ajax which have stood the test of time were on show on occasions this evening, not least for the goal. Similar performances in their remaining group games against Barcelona and AC Milan may not be enough, but it was a must-win game for them this evening and they did just about enough to pick up all three points.
Neil Lennon's side just don't seem to be able to perform away from home in Europe. They have now lost 24 of their 26 away games in the group stages and beyond of this competition, and it was clear to see why this evening. Their general standard of play wasn't too bad, but they seemed to lack a cutting edge in the final third, with Ajax keeper Jasper Cillessen having little to do all night. They were not well-beaten by Ajax but they deserved to lose the game and now face an uphill struggle to even qualify for the Europa League after Christmas.
Sports Mole's man of the match
Lasse Schone: Schone was the most constant threat to the Celtic defence this evening as they struggled to pick up his late darts into the area. The Dane was ultimately the difference between the two sides, scoring the winning goal with a composed finish after linking up with Thulani Serero on the edge of the box. It was a lovely interchange between the pair, and Schone in particular deserves a lot of praise for it.
It was a relatively gaffe-free game, but Fraser Forster did have a heart-stopping moment in the second half when he completely misjudged a corner into the box. The England hopeful failed to stop it curling over his own head and towards the top corner, and he was only saved from the embarrassment of conceding direct from a corner by Emilio Izaguirre, who cleared it off the line.
It was a lenient display by Deniz Aytekin of Germany. He let a lot of things go in the first half, giving free kicks but keeping his cards in his pocket when a couple of the challenges deserved cautions. He made a couple of odd decisions here and there, but overall he didn't have much to do and controlled the game well.
Ajax: Ajax will be looking to get their season back on track when they face NEC in the Eredivise on Sunday.
Celtic: Celtic, meanwhile, travel to face Ross County on Saturday as they look to maintain their unbeaten start to the season.