In contrast to City's woes, modest Spanish side Malaga made light of it being their first appearance in the competition by qualifying for the knockout phase - after a 1-1 draw with AC Milan - and were joined by 2004 winners Porto, who drew 0-0 away at Dynamo Kiev.
City's hopes hang by a thread as with two matches to go they are five points adrift of second-placed Real Madrid, who had to twice come from behind to draw 2-2 at home to Borussia Dortmund, who top the group on eight points.
Ajax are third on four points and City manager Roberto Mancini believes his side are all but out..
"We conceded two stupid goals. In a very important game like tonight, you can't concede two goals from corners. Now we can do nothing, it's finished."
City, having taken just one point from their opening three fixtures - including a shock loss in Amsterdam - looked out for the count after an atrocious start.
Their defence was all at sea as Siem De Yong netted a double in the opening 19 minutes.
But an acrobatic effort from Yaya Toure put the English champions back in the contest and then Sergio Aguero fired home a leveller midway through the second period.
Drama all around
There was no less drama at the Bernabeu where Borussia Dortmund so nearly managed what City had come within minutes of achieving in their opening pool match - handing Real Madrid a defeat on their home patch before the Spanish snatched a 2-2 draw.
Real, nine-times champions, found themselves behind on 28 minutes after Robert Lewandowski found Marco Reus, who eluded Alvaro Arbeloa before firing a right-footed shot into the top corner past Spain keeper Iker Casillas.
Jose Mourinho's side swiftly drew level thanks to a towering header from Pepe. The Spanish looked finished after Mario Goetze's attentions forced Arbeloa into sending the ball past Casillas but Meszut Oezil curled in an 89th-minute freekick to bag a point.
Elsewhere, Arsenal were held 2-2 away at Schalke 04 after letting slip a 2-0 lead, which leaves them a point adrift of the German side.
With Olympiakos seeing off eliminated French champions Montpellier 3-1 Arsenal have only a one-point advantage over the third-placed Greeks with two games remaining.
Malaga took the lead against seven-time champions AC Milan, Eliseu putting the visitors ahead against the run of play before Alexandre Pato levelled for a draw which leaves Malaga on ten points from four games, five ahead of Milan. Milan have work to do as they stand only one point ahead of Anderlecht after the Belgians edged out Zenit St. Petersburg 1-0. Porto, champions under Mourinho in 2004 march on having notched up ten poiints already after a goalless stalemate at Dynamo Kiev.
Group A rivals Paris Saint Germain are well-placed to reach the last 16 alongside the Portuguese after thrashing Dinamo Zagreb 4-0.