City suffered a potentially fatal blow to their hopes of retaining their Barclays Premier League title as they were beaten 2-0 by their bogey side at Goodison Park.
Leon Osman and Nikica Jelavic struck as the Toffees responded to their own critics after their dismal FA Cup quarter-final loss to Wigan by beating City for a sixth time in eight meetings.
They had another goal from Kevin Mirallas harshly ruled out for offside and Marouane Fellaini was denied a penalty in the first half.
City could have had their own penalty with the score still 1-0 in the closing minutes but Everton held out despite playing the last half-hour with 10 men after Steven Pienaar was sent off.
Manager Roberto Mancini did not appear for post-match interviews, leaving first-team coach Platt to answer questions.
When asked why Mancini did not speak, Platt told Sky Sports: "He is a bit angry, as you can imagine.
"He is just taking stock of the situation and calming down. He doesn't want to come out here and say anything that might get him in trouble."
When asked what Mancini might be angry about, Platt said: "Everything in general, our performance - we weren't really at it throughout the game.
"We got outworked by Everton. When you do have moments when you can get back into it, it doesn't go for you."
Platt later attempted to clarify his remark about being "outworked" when he spoke to the rest of the press.
He said: "If it has come across like that it certainly wasn't meant.
"You know they are going to make you work. If you look at the stats - we have probably covered the same amount of distance, shown the same intensity."
City could have had a penalty when a Carlos Tevez shot struck Fellaini's hand inside the box. Referee Lee Probert gave a free-kick on the edge of the area and Aleksandar Kolarov fired into the wall.
It was suggested the ball may have hit Osman's hand before Fellaini's, which might explain Probert's decision, but Platt was unsure.
He said: "I didn't see whether it brushed Osman's hand but I don't think that would have been given because it has not stopped the flight of the ball.
"I think it has been given for the Fellaini handball. Is it intentional? No, but having given it, it is two or three yards inside the box.
"It is a decision that has gone against us."
Platt intends to keep fighting even though the title now looks even more likely to be reclaimed by Manchester United.
He said: "We have an obligation to win football matches between now and the end of the season, regardless of whether there is a points advantage or it is mathematically impossible, which it isn't at this moment."
For Everton, the performance and result was a perfect response after a week of soul-searching following the Wigan loss.
It also ensured manager David Moyes would have something pleasing to mark the 11th anniversary of his appointment with.
Moyes said: "I would have been disappointed if we hadn't responded, and surprised if we hadn't.
"The players played great. We were playing the champions of England and I thought we gave them a right good run for their money.
"What you saw is what Everton do. I thought we played really well in the first half and I thought in the second half, when we were down to 10 men, we did what Everton did 11 years ago when I took over.
"We were resilient, hard to beat, made sure the opposition didn't score, threw ourselves on the line for the team.
"We were rubbish last week but we have been good all season and you are watching a really good Everton side.
"It is just to make sure people realise we have not gone away. We are still there. I think people last week thought we had gone away."
Moyes was far from impressed by the sending-off of Pienaar for fouls on Gareth Barry and Javi Garcia.
He said: "I think the sending-off is a terrible decision. The boy rolls about three times and there is hardly any contact. I thought it was the wrong decision."