Lasse Schone, who had scored after coming on as substitute in the 2-1 defeat at Celtic Park, grabbed the winner with a well-taken goal in the 51st minute to sink the Scottish champions - who were far from their best in what was a mostly scrappy game.
Ajax move above them on four points with Barcelona having qualified on 10 points following their 3-1 win over AC Milan at the Nou Camp.
The Italian side visit Celtic Park later this month before the Hoops travel to the Nou Camp in December to play Barcelona.
While there are still various permutations available, Celtic will be out of the competition if they lose to Milan in their next game.
The Hoops boss said: "It was an opportunity lost tonight but all is not lost.
"It is imperative that we beat Milan at Celtic Park if we have are to have any chance of qualification. But we will have to play better than on the first-half evidence tonight.
"Our play in general was not like us tonight.
"We were very flat in the first half and I think some players will have regrets because they were nowhere near their best and at this level you can't have that.
"We missed some decent opportunities, particularly the one when James Forrest one when were two on one.
"You were thinking 'goal' and that would have got us back in the game.
"We could have done better with the final ball and that has been a concern for the last few weeks.
"Goals change games and from their point of view it was good but from our point of view it was a poor goal.
"When Schone scored we were just getting a foothold in the game.
"But it could all flip round if we beat AC Milan and Barcelona come here and take something off Ajax."
Ajax coach Frank De Boer refused to write off Celtic's chances of qualifying out of the group.
He said: "Celtic still have a chance, they are still only two points behind Milan.
"There was more pressure on us after the game in Scotland so we are relieved.
"It is still wide open and we have still to go to the San Siro.
"We also have to play against Barcelona and a point will be a bonus point.
"You seen from Celtic last year (against Barcelona) that anything is possible.
"They are already qualified and we have nothing to lose."
The former Rangers player insists his side deserved to win but could have made it easier for themselves by getting a second.
He said: "We know if we play like we have to play, and the way we used to play, we know we can win and we deserved to win.
"The first half was very good, we had some good moments and we were very happy to score to make it 1-0 but we forgot to score the second which would have made it easier for us.
"But we looked a bit tired and maybe that is why."