Danish international Schone's goal came at the end of a series of swift interchanges of short passes involving five different Ajax players as they picked open the Scottish club's defence to break the deadlock in the 51st minute.
Defeat for Celtic left them bottom with three points and matches to come at home to Milan this month and away at leaders Barcelona in December.
The Glasgow club also wasted chances to at least share the spoils when they twice came close on the counter-attack in a match played at high tempo between two evenly-matched sides.
Beram Kayal shot wildly over at the end of a long surge upfield in the 50th minute that saw the home defenders backing off and being pulled wide by other players and a gap opening up for the Celtic midfielder who wasted his chance.
Seven minutes later James Forrest and Anthony Stokes were two up against a solitary Ajax defender on a swift counter attack but Forrest fluffed his attempted pass to Stokes who would have needed just to place the ball into the net for the equaliser.
Ajax had an in-swinging corner cleared off the line by fullback Emilio Izaguirre.
(Reporting by Mark Gleeson; Editing by Ed Osmond)
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