Stefano Denswil forced Fraser Forster into an excellent diving save to his left with a curling free-kick early on, while the Celtic keeper dived to his right to deny Siem de Jong.
But the Dutch outfit took the lead on 51 minutes when Schone, who had scored after coming on as substitute in the 2-1 defeat at Celtic Park, finished off a great team move.
His goal came just after Celtic midfielder Beram Kayal had missed a good chance for the visitors and before Anthony Stokes failed with an opportunity to level.
Denswil then headed against the post late on as Frank de Boar's team held on to win and leapfrog Celtic into third in the group.
The Hoops, whose Dutch defender Virgil van Dijk was immense, remain on three points with two fixtures remaining. Ajax move onto four points with Barcelona having qualified following their 3-1 win over AC Milan at the Nou Camp in the group's other game.
AC Milan visit Celtic Park later this month before the Hoops travel to the Nou Camp in December to play Barca.
Celtic were boosted by the return of Georgios Samaras and Kris Commons who recovered from respective knee and hamstring injuries.
Skipper Scott Brown and Nir Biton missed out through suspension but Mikael Lustig shrugged off an on-going hip problem to take his place at right-back with Kayal coming into midfield.
Ajax coach de Boer made three changes from the team that lost 2-1 in Glasgow - Nicolai Boilesen, Davy Klaassen and Schone came in for Lucas Andersen, Christian Poulsen and Viktor Fischer, all of whom started on the bench.
In a reshuffled line-up which smacked a little of desperation, De Boer moved Daley Blind from left-back into midfield and captain de Jong from midfield into a forward position.
The decision to close the stadium roof had been taken during the day which meant the decibel level was raised to produce a great atmosphere.
The home side started with purpose and in the sixth minute Forster made a good save from Denswil's 25-yard free-kick, after Commons had fouled Blind, with the corner coming to nothing.
In the 16th minute Forster came to the rescue again when he pushed away a downward header from de Jong, who had escaped Efe Ambrose to get on to the end of a van Rhijn cross.
There was an Ajax claim for a penalty when Ricardo van Rhijn's cross struck van Dijk but referee Deniz Aytekin was unimpressed.
Four minutes into the second half, Kayal was allowed to run 50 yards to the edge of the Ajax penalty area but he blazed his drive high over.
Celtic were made to pay moments later when Schone slipped the ball past Forster from 12 yards after working a one-two with Thulani Serero at the edge of the Hoops box.
With that, the game opened up further and Stokes had a chance in the 54th minute when he was set up by Commons but his low drive was dealt with by Jasper Cillesen.
At the other end, Schone's in-swinging corner from the left was headed off the line by Emilio Izaguirre with Forster beaten.
In the 57th minute James Forrest should have picked out Stokes as he raced down the right but managed to hit Joel Veltman, the only Ajax outfield player back.
Lennon's side had another opportunity in the 67th minute when Klassen was penalised for blocking Commons' pass 25 yards from goal but he drove the ball straight to a grateful Cillesen.
Celtic's Dutch forward Derk Boeriggter replaced Stokes, Joe Ledley came on for Kayal and Teemu Pukki took over from Commons as Lennon tried another route.
Both sides continued to spurn chances, with Forster making a save from de Jong with five minutes remaining before the unmarked Denswil headed the corner off the post and behind as Ajax held on to claim a crucial win.