Cameron Norman's bullet header and a debut goal for Calum Kavanagh saw off 10-man Swindon at Rodney Parade.
The game was always against Swindon after losing Rushian Hepburn-Murphy to a 15th-minute red card, and Norman's header in the 54th minute ensured County made full advantage of playing the majority of the match with a man extra.
And with 11 minutes left, County made it 2-0 as Kavanagh tapped into an empty net to put the game to bed.
County had the first chance of the game, but Swindon keeper Sol Brynn was equal to Priestley Farquharson's header following Aaron Lewis' free-kick after two minutes.
Charlie Austin had a good opening for Swindon soon after, but the Town captain saw his strike from inside the six-yard box deflected behind for a corner.
Newport managed to clear said corner, but only as far as former loanee Jake Cain who fizzed a half-volley from the edge of the box over Nick Townsend's crossbar.
Disaster struck for Swindon on the quarter-hour mark though when Swindon forward Hepburn-Murphy was shown a straight red card for pushing Norman in the face after being aggravated during a scuffle on the right touchline.
Graham Coughlan's men tried to make the one-man advantage count immediately, but Adam Lewis could only hit his half-volley from the left of the area wide of the near post after a ball in from the right.
Norman would then get in the thick of the action again, as he first hit a low swerving volley into the arms of Brynn from the edge of the box, and a minute later he crossed for Harry Charsley who headed over the bar.
Brynn was kept busy and was needed to tip a Mickey Demetriou cross from the left over the bar that would've dipped in if not for Brynn taking decisive action.
And on the half-hour mark Charsley shot over the bar for a second time after good link-up play from County ended with fellow new boy Charlie McNeill laying the ball off for Charsley.
County had the first chance of the second half, but again it was an easy save for Brynn as Aaron Lewis' header from inside the area was straight at him after Charsley provided the cross.
Newport did get the lead in the 54th minute though when Adam Lewis' cross from the left was nodded home ferociously by opposite wing-back Norman after he ghosted in at the back post.
Nathan Moriah-Welsh was introduced as a second-half sub and had a good effort from the edge of the box saved by Brynn, who was equal to his half-volley from the D, as Newport pushed for the second.
It arrived in the 78th minute when Aaron Lewis broke the offside trap superbly which set up a two-on-one for himself and Kavanagh.
Lewis was unselfish and squared for Kavanagh which took Brynn out the picture and Kavanagh had all the time in the world to compose himself and tap into an empty net for 2-0 and a perfect start to life at County.
Swindon rarely threatened all game, but Townsend was equal to Luke Jephcott's effort to keep it 2-0 in the final five minutes.
Tomi Adeloye did pull one back for Swindon in second-half stoppage time, but Newport saw the game out to claim a vital three points.
Omar Bogle could have even made it 3-1 in the dying embers, but his effort came back off the post.
Newport County: Nick Townsend, James Clarke, Priestley Farquharson, Mickey Demetriou (C), Cameron Norman, Adam Lewis, Scot Bennett, Aaron Lewis (Omar Bogle '85), Harry Charsley (Nathan Moriah-Welsh '70), Offrande Zanzala (Will Evans '80), Charlie McNeill (Calum Kavanagh '70)
Subs not used: Joe Day, Matt Baker, James Waite
Swindon Town: Sol Brynn, Remeao Hutton, Ciaran Brennan, Frazer Blake-Tracy, Ellis Iandolo, Dylan Kadji (Luke Jephcott '68), Jake Cain, Jacob Wakeling, Jonny Williams (Ronan Darcy '59), Rushian Hepburn-Murphy, Charlie Austin (C) (Tomi Adeloye '89)
Subs not used: Jack Copland, Harrison Minturn, Ricky Aguiar