Their team spirit was evident in south Wales as they dominated the Exiles at Rodney Parade.
The Shakers are now four points off the automatic promotion places and have a game in hand on third-placed Wycombe.
Ryan Lowe flicked a shot from Chris Hussey past Newport goalkeeper Joe Day to snatch a first-half lead.
Adam El-Abd almost threw his side’s advantage away when he stabbed a Miles Storey cross onto his own post.
But Nardiello nipped in to seal the points at the death.
Flitcroft said: “I’m absolutely delighted, the Newport boys really play for their manager and are driven on by the coaching staff.
“In the build-up all the talk was about what they are going to do to us and how they were going to spoil the party and take the game to us.
“But we didn’t allow them to take the game to us and that’s what I’m really pleased with.
“We’ve worked on it all week and made sure we dominated the ball in the first half, and the second-half performance pleased me more than anything.
“We’re making progress in every aspect of the football club and moving to (former Manchester City training ground) Carrington is assisting in that.
“It’s certainly helped and preparation has been key.
“We’ll have a session in the ice baths when we get back to Bury this evening, it’s never ending.
“That’s what sacrifice is. It’s about committing day in day out for each other and that was evident today.”
The result sees Newport slip out of the play-off places following their second successive defeat.
Manager Jimmy Dack said: “I think we had a few early chances that we needed to take, but we’ve come up against a good side today in Bury.
“That’s the disappointing part about it, we have to start getting that first goal. We don’t always have the quality in the final third when we are chasing games and we lacked it today.
“We were beaten by the better team without a doubt and I am not going to dress it up in any other way.
“They fully deserved to win and are a good team who we gave it a good go against, but just fell a bit short.
“But there is four games still to go and in terms of our chances of getting in the play-offs, we feel we are really still right in it and it will go down to the wire.”