Tuesday's result left the English champions five points below second-place Real Madrid in Group D with only two points from four games, and although wins in their last two matches could take them through, Mancini feels it is too late.
"I don't think we have a chance," said the City coach, who went onto the pitch to confront referee Peter Rasmussen at the final whistle.
"Eight points if we win both games. If we do that and go through, we can win the Champions League! It would be our destiny.
"But I believe in my team always, even maybe sometimes when there are mistakes."
City had two penalty appeals turned down, one for an early challenge on Yaya Toure and the other in injury time after Mario Balotelli appeared to be manhandled in the visitors' box.
They were also controversially denied an 88th-minute equaliser when Aleksandar Kolarov was ruled to have strayed offside before crossing for Sergio Aguero to tap home.
Mancini confessed that he had not seen the incident involving Balotelli, but he admitted haranguing the match official about the decision not to let Aguero's goal stand.
"I said it was a goal," he said in his post-match press conference.
"I didn't see the last bit with Mario that was a penalty. The referee and his staff were really poor tonight."
Mancini also dismissed reports he had threatened a television cameraman in the immediate aftermath of the match.
"The cameraman wanted to film me and follow me from the pitch," said the Italian. "I said it's finished. It's not a film about me -- it's a game."
Mancini's opposite number, Frank de Boer, said he thought the officials had been correct to rule out Aguero's goal.
"I thought immediately that Kolarov was offside," he said.
"I looked at the referee and the assistant referee, and they were very confident and had no doubts about it.
"For the other one (Balotelli's penalty appeal), I couldn't see from the bench. But sometimes you have good luck with referees, and sometimes you have bad luck."
Although the decisions of Danish official Rasmussen gave City reason to feel aggrieved, they only had themselves to blame for some inept defending in the first half.
Ajax captain Siem de Jong capitalised twice, scoring from close range in the 10th minute and then heading in at the near post in the 17th minute after City were twice caught napping at corners.
However, Toure's fine chest-and-volley mid-way through the first half brought the hosts back into the game and Aguero equalised 16 minutes from time.
It leaves City on the verge of elimination in the group phase for the second year in a row, and Mancini said his side had been made to pay for previous failings in the 3-2 loss at Real Madrid and the 3-1 defeat at Ajax.
"We conceded six goals against Real Madrid and Ajax," he said.
"At 2-1 at the Bernabeu (against Madrid), you have a chance to take at least a point, not lose the game.
"Against Ajax (in Amsterdam), we didn't play well. We were 1-0 up and didn't take our chance to score a second goal. Maybe it's not our moment in the Champions League."
City's campaign will be over if they fail to win at home to Madrid in their next game on November 21, but Ajax will approach the visit of leaders Borussia Dortmund with renewed hope of crashing into the qualifying places.
Source: AFP