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 the Dutch side is 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 favor of Davy Klaassen, while Celtic welcomed back key man Kris Commons from a hamstring injury.
Off color 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.
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 back-heel past Forster.
Anthony Stokes drew a first save of the night from Jasper Cillessen as Celtic sought a quick response, but it was almost 2-0 down in the 56th minute when Forster misjudged Schone's in-swinging corner and Emilio Izaguirre headed off the line.
James Forrest then failed to make the most of a two-on-one counterattack situation in Celtic's favor 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 hosts AC Milan next time out before closing its group stage campaign at Camp Nou against Barcelona.