Celtic's Europa League hopes were ended for the season as Ajax substitute Vaclav Cerny scored a late winner for the visitors at Parkhead.
Hoops returning midfielder Callum McGregor, only starting due to a midfield crisis, gave the home side the lead and some hope in the third minute with a drive.
The Dutch side recovered and by the time Arkadiusz Milik levelled in the 21st minute they were bossing most of a wide open match but Celtic reasserted themselves only to be stung in the 88th minute with Cerny's breakway winner to win 2-1, although Fenerbahce's win against already-qualified Molde made the result academic.
Celtic remain bottom of Group A with two points with their final match against Fenernbahce in Istanbul next month a dead rubber
Ronny Deila had been left with something of a midfield crisis after it was revealed on Wednesday that skipper Scott Brown could be out for up to three months with a knee injury.
With Kris Commons also suffering from a hamstring complaint and illness, and Nir Bitton and Stefan Johansen suspended, the Norwegian had to dig deep into his squad.
McGregor was handed a rare start with Gary Mackay-Steven also coming in and 18-year-old left-back Kieran Tierney replacing the more experienced Emilio Izaguirre.
The Hoops got off to a dream start, however, when McGregor took a pass from Mackay-Steven and edged inside on to his right foot before despatching a shot from 20 yards past Ajax keeper Jasper Cillesen.
The visitors were clearly rattled but they slowly composed themselves.
Viktor Fischer had the ball in the net in the 11th minute with a thunderous drive but the offside flag was already up.
The Dutch side settled down further and began to move the ball around the pitch with increasing confidence and there was no surprise when their equaliser arrived, although it contained an element of luck.
Fischer took advantage of a deflected strike from Lasse Schone which landed in his path inside the box and although Celtic keeper Craig Gordon beat away his close-range strike, he headed the rebound back across goal where Polish striker Milik knocked it over the line.
Celtic, trying to get their shape right, retreated to the edge of their box as revitalised Ajax probed and prodded and just over the half-hour mark winger Amin Younes curled a shot just over the bar from 16 yards.
Both sides were still going at each other as the interval whistle sounded and the promise of goals at either end remained in the second-half, although both defences had tightened up.
In the 64th minute Charlie Mulgrew, back for the first time in over two months after recovering from a thigh injury, replaced Tom Rogic, just after Cillesen had made a save from James Forrest's low drive and seconds before McGregor's deflected drive from 25 yards had hit the bar and gone over for a corner which came to nothing.
As the match became increasingly fiesty - Ajax were putting in some hefty challenges - Mackay-Steven was replaced by Scott Allan, who moments later put Leigh Griffiths through only for Cillesen to block his shot for a corner from which the visitors were fortunate to escape.
Izaguirre replaced Tierney in the 77th minute by which time Celtic had the bit between their teeth and Ajax looked tired.
With five minutes remaining, after Cillesen had made a comfortable save from Griffiths, the impressive Younes crashed a shot against Gordon's post before Ajax broke and Cerny, on for Fischer, drove in number two to sink the home side.
TWEET OF THE MATCH
?@mj88live: "Gutted. Where do we go from this campaign...? ?? #Celtic #europaleague" - Olympic silver medalist swimmer Michael Jamieson doe snot take Celtic's elimination well. https://twitter.com/mj88live/status/669998396988653568
CRAIG GORDON 6
MIKAEL LUSTIG 6
DEDRYCK BOYATA 5
JOZO SIMUNOVIC 7
KIERAN TIERNEY 6
TOM ROGIC 5
STUART ARMSTRONG 6
JAMES FORREST 7
CALLUM MCGREGOR 7
GARY MACKAY-STEVEN 5
LEIGH GRIFFITHS 7
CHARLIE MULGREW (on for Tom Rogic, 66) 6
SCOTT ALLAN (on for Mackay-Steven, 71) 6
EMILIO IZAGUIRRE (on for Tierneyt, 77) 5
MOMENT OF THE MATCH
Callum McGregor's opener gave Celtic fans some hope of a great European night but once again in Europe this season the Hoops ran out of steam.
VIEW FROM THE BENCH
After two points from four Group A matches Celtic manager Ronny Deila badly needed the win, not just for Europa League hopes but also his credibility on the European scene. But now he has to concentrate on the domestic scene with his reputation suffering another blow.
Ajax boss Frank de Boer had lost 2-1 at Celtic Park in the Champions League in 2013 so he will be glad of revenge.
MOAN OF THE MATCH
From Celtic's point of view it will be allowing a lead in Europe to slip once again and the almost customary two goals were lost.