The match finished 4 - 3 to Peterborough in a see-saw battle that saw City lead, go behind, lead again before the final blow in stoppage time.
Mackay said: "The referee made some incredible decisions that undoubtedly influenced the game.
"If the referee wants to give Peterborough a penalty for handball, he should have given us one earlier.
"We also had a goal disallowed that would have put us 2-0 up early on when we haven't had a player go offside at any point in the move.
"I asked the referee to speak to him calmly for a couple of minutes but he enforced the 30-minute rule on me. He is the first official to do that.
"It was a game of so many talking points and one we definitely did not deserve to lose."
Posh boss Darren Ferguson had no such complaints as his side won an 87th minute penalty and then scored a 92nd minute winner when it could have been called offside.
"It is the first time we have managed back-to-back wins this season and that is a big achievement," Ferguson said.
"It was a terrific game and a great way to win. No defender is catching Paul Taylor when he gets going and he showed terrific composure to produce a great finish.
"If Lionel Messi had done that in La Liga everyone would have been drooling over it.
"You don't win matches like that if you don't possess bags of character. All the ability in the world is no use without the heart and fight we showed.
"We also know that if we get another win on Saturday we'll go second in the league and that's a brilliant place to be."