Ajax skipper Siem de Jong scored twice from two corners in the opening 17 minutes to leave the Eastlands outfit on the brink of elimination from the Champions League.
However, Yaya Touré pulled a goal back with a superb close-range finish before Sergio Aguero netted in the 74th minute to give City hope of a comeback.
The Argentina international was denied a dramatic winner due to a controversial offside decision, while the officials failed to award a 93rd-minute penalty when Mario Balotelli was felled in the area.
“The referee and linesman were really poor,” said Mancini. “Yes [we should have had a penalty] and a goal.”
“I told him it was a goal and not offside.”
He added: “We played well. We conceded two goals from two corners and the game was changed.
“But in the second half we played very well and we had a lot of chances. We scored three goals and had a penalty.”
City captain Vincent Kompany admitted the Citizens were particularly frustrated to concede two goals from set-pieces after keeping Ajax at bay in open play.
“It’s very important you defend as a team from corners and we let ourselves down on these two phases,” said Kompany.
“Especially, when you look at the game, they didn’t create that many chances unlike in Amsterdam.
“It was just a case of us scoring and those two goals were not good.”
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