The Exiles imploded when 4-1 down on home soil to finish with nine men after red cards for Ciaran Brennan and Kyle Jameson.
The Valiants were already in control of the game as they showed their quality in the bid to bounce back from League One relegation.
They led 2-1 at the break courtesy of a brace by Jayden Stockley either side of an Aaron Wildig header.
Ethan Chislett and Lorent Tolaj struck early in the second half before the County dismissals, Brennan for a spat with Ben Garrity after grappling and Jameson for kicking the ball into the away end.
Vale should have added to their tally but Moore, who was Brennan’s boss at Sheffield Wednesday, was content with the win after his side headed to Wales on the back of losses to Barrow and Doncaster.
“We came to get the right result, and we got that. It was a good performance, a solid performance, and I thought we saw what the team is capable of in glimpses,” said the former centre-back.
“I thought we were excellent from the word go. We made a fast start and it was really difficult for Newport to get any momentum.
“Credit to the players, even when it went to 1-1, we regained our composure and deservedly won the game.
“Even with the two sendings off, our boys kept their heads and made sure that we kept 11 players on the pitch.
“When others are losing their heads, you have to make sure you don’t get caught up with the atmosphere, and I thought the players did that really well.
“I hope the Vale fan who got hit by the ball is OK. It was a solid away performance and we got the three points that we wanted.”
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