The Bhoys are bottom of Group H following defeat in Amsterdam and have to get results against AC Milan and Barcelona if they are to seal a route to the knockout stages
Lasse Schone, who scored a late consolation as the Scottish champions claimed a 2-1 Group H win at a raucous Celtic Park two weeks ago, got revenge in Amsterdam by scoring six minutes into the second half.
Another frustrating evening appeared to be in store for Frank de Boer's men after Celtic repelled wave after wave of first-half attacks, but Schone claimed the breakthrough with a wonderfully crafted effort.
A second is often necessary for Ajax but a first Champions League clean sheet in 11 attempts means they are now a point behind second-placed Milan with two games remaining, while the Bhoys are bottom and must face Milan and Barcelona.
De Boer dropped winger Viktor Fischer in favour of Davy Klaassen, while Celtic welcomed back key man Kris Commons from a hamstring injury.
Off colour both at home and abroad so far this season, the defending Dutch champions began what had been billed as a must-win assignment on the front foot, and Celtic goalkeeper Fraser Forster was forced to make a sharp save from Stefano Denswil's free kick.
Forster then demonstrated again just why he is being spoken about as a potential first-choice for England with a magnificent reaction stop to claw away Siem de Jong's 17th-minute header from Ricardo van Rhijn's cross.
Georgios Samaras neglected his defensive duties down the Celtic left on that occasion and the Greece forward was booked for a foul on Thulani Serero two minutes later, Forster holding on well as Denswil's resulting zipped attempt through the wall.
More stubborn defensive efforts from the visitors meant Ajax departed for half-time frustrated.
Beram Kayal spurned a rare Celtic opportunity in the 50th minute, blazing wildly over at the end of a driving run, and he was soon ruing his profligacy.
Having previously failed to find an incisive pass, Ajax strung together a slick one-touch move to break the deadlock - Schone exchanging passes with De Jong and Thulani Serero before slotting the latter's backheel past Forster.
Anthony Stokes drew a first save of the night from Jasper Cillessen as Celtic sought a quick response but they were almost 2-0 down in the 56th minute when Forster misjudged Schone's inswinging corner and Emilio Izaguirre headed off the line.
James Forrest then failed to make the most of a two-on-one counter-attack situation in Celtic's favour before Serero forced another last-ditch block from Izaguirre inside the Celtic box.
Forster denied De Jong at the end of a bustling run and Ajax ended the strongest, with Denswil heading against the post.
Celtic host AC Milan next time out before closing their group stage campaign at Camp Nou against Barcelona.