Tyler Walker gave the visitors the lead just before the interval but Patrick Bauer and Emil Riis struck twice in eight minutes to turn the game in PNE's favour.
It was the first win in the Championship since beating Swansea City at the end of August.
PNE head coach McAvoy: "I didn't think there was much in it in the first half.
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"We were a bit slack and got done in transition for the goal.
"Coventry are a good team, they open the pitch up well. We had spoken long and hard about how to go about our business.
"We spoke to them at half-time and said we need to be a bit more aggressive, get up against them a bit more and in their faces, take it to them and get into the final third.
"I thought that the players' application, hunger and desire was excellent. I felt they showed that."
McAvoy had come in for plenty of criticism of late following a run of six draws and a defeat in the league.
That had intensified in the wake of last Saturday's drab 0-0 draw with Derby County.
McAvoy said: "There is criticism and there is being over-criticised.
"We have been in charge 24 games, of which we have won 11 and drawn eight. There are 19 results there which in my book haven't been a bad return.
"I know latterly we have drawn too many games but that has not been through the lack of trying.
"I don't wear it that we should have lost all those games, that is nonsense - everyone knows it is a nonsense.
"I'm glad we had the guts to have stuck to our system tonight, because there has been a lot of pressure to change it.
"The 3-5-2 has in the main served us well, Josh Earl tonight as a wing-back was magnificent - his second-half performance was superb.
"I thought we got DJ and Alan Browne into pockets of space in the second half, which we didn't do enough in the first half.
"Once we did that in the second half, some of the football was excellent.
"I was delighted for them, I know they are a good group."
Bauer's goal was his second of the season, with Riis now having netted nine in league and cup.
Victory came on the night club captain Alan Browne made his 300th appearance for the club - only the 27th player to reach that milestone.
McAvoy said: "Patrick Bauer has been magnificent for us, he's come back from a horrific injury, he's a leader.
"It was Alan Browne's 300th appearance. That was one of the things we spoke about, let's not disappoint him - let's give him something to remember.
"Some people never reach that in their playing career so for Alan I'm delighted. He was magnificent and worked his socks off
"Emil is a young man learning his trade, He has been involved in every game this season because I know he is a goal threat."
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