Leeds, 53 places higher on the league ladder than Newport, led 1-0 for most of a pulsating third-round tie at Rodney Parade.
But a Conor Shaugnessy own goal and substitute Shawn McCoulsky’s towering header put County in the fourth round for the first time since the 1978-79 season.
A grim afternoon for Leeds got even worse when Samuel Saiz was sent off in stoppage time, the Spaniard banished after allegedly spitting at County midfielder Robbie Willmott.
“I thought we were outstanding throughout, but I knew we were going to put a performance in because they never let me down,” said Flynn, who saved Newport from dropping out of the English Football League last season.
“They are an outstanding group and deserve all the credit.
“Being a Liverpool fan, I’d love us to play at Anfield in the next round.
“But, really, it’s whatever makes the club the most money.
“Hopefully we can get one of the big clubs away – the money will be huge for this football club.”
Newport trailed to Gaetano Berardi’s ninth-minute strike – which possibly brushed off the heel of team-mate Pierre-Michel Lasogga – but created chances throughout.
They finally took advantage of a Leeds side showing nine changes from their New Year’s Day draw at Nottingham Forest when Shaugnessy sliced Frank Nouble’s cross into his own net 14 minutes from time.
McCoulsky, on loan from Bristol City, then headed home from a last-minute corner to send Rodney Parade wild.
“I was a little bit more confident when I saw their team, but they are Championship players and you’ve still got to beat them,” Flynn said.
“I was confident anyway with the nature of them coming down here to our place.
“It’s not their home comforts and I knew we would make it difficult for them.”
Leeds manager Thomas Christiansen denied he had disrespected the FA Cup by making so many changes to his side.
“If you look, every player we had in the first XI today is from the first team or has possibilities in the first XI,” Christiansen said.
“For this we expect more and it should be enough to pass to the next round.
“Against Burnley (in the Carabao Cup) I also made 10 changes and we had the perfect performance.
“It was the same against Leicester, we also made nine or 10 changes and did very well.
“So why not today? Why is it different? They know what the FA Cup means and I know it.”
On the Saiz spitting allegation, Christiansen added: “I do not know, but if that’s right why he got the red card then it’s a problem. But I believe he didn’t do it.”
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