City fought back from 2-0 down to 2-2 in a gripping Group D encounter before having a goal disallowed and being denied a penalty in the closing moments.
Victory would have kept their hopes of reaching the knockout stages in their hands but the result all but eliminates them from the competition prematurely for a second successive year.
Mancini went on to the field to confront referee Peter Rasmussen after the final whistle, with his main gripe being the 88th-minute goal Sergio Aguero had ruled out for offside by Aleksandar Kolarov.
City's players also complained after Mario Balotelli was denied a spot-kick for a shirt pull in injury time because Rasmussen blew the final whistle.
Asked what he said to the official, Mancini said: "I said it was a goal. I didn't see the last situation with Mario that was a penalty. I said it was a goal.
"The referee and his staff were really poor tonight."
City remain bottom of the group with just two points from four games.
They must now beat Real Madrid and Borussia Dortmund in the final two matches to have any hope of progressing, but other results will still have to go their way.
Their underwhelming campaign went from bad to worse as Siem de Jong scored twice for the Dutch side inside the opening 17 minutes.
Yaya Toure pulled one back with a fine volley and Aguero levelled 16 minutes from time but, despite the frantic and unlucky finish, it was not enough.
Mancini said: "I think that we scored three goals.
"We conceded two stupid goals. In a game as important as tonight you can't concede two goals from corners.
"After that moment we played well. We scored the first goal and were fantastic in the second half.
"But now, we can do nothing, it is finished."
Ajax are third in the group and still in with a realistic shot of qualifying, but coach Frank de Boer was disappointed his team did not hold onto their lead.
The former Holland defender said: "It is a good sign that the players in the dressing room are disappointed about the result.
"The feeling was that we could make a big step forward towards the next phase - the Europa League or the Champions League - and put City behind us.
"We gave the goals away cheaply. It was very disappointing for us.
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