Town led twice through a Carlos Mendes Gomes stunner and a Tom Lockyer goal, but both times Newport pegged them back with efforts from Lewis Collins and Chanka Zimba before James Waite grabbed a winner with 14 minutes to go.
Matt Macey was named in goal for his second Luton debut after returning to Kenilworth Road this summer, while Tom Lockyer captained the side in a back three alongside Henri Lansbury and Reece Burke, whilst there was a competitive Luton debut for Louie Watson in midfield.
Cauley Woodrow started and in doing so made his first competitive Luton start across his two spells at the club, supporting attackers Cameron Jerome and Admiral Muskwe.
Seventeen-year-old forward Zack Nelson is amongst the substitutes, as well as Development Squad regulars Josh Williams, Casey Pettit and striker Josh Allen.
The Hatters started brightly, Muskwe and Berry registering shots on target in the opening ten minutes, with Pereira sending a trio of dangerous looking crosses into the Exiles' box without reward before Watson curled just wide from 20 yards.
Berry saw an attempted curler pushed away by Joe Day, Jerome got his head on a Lansbury free-kick but couldn't direct it on target,then nodded down for Muskwe, whose attempt to nick the ball over the County keeper resulted in it ricocheting back off his legs for a goal-kick.
It took a moment of brilliance from Mendes Gomes to finally break the deadlock on the half-hour, when he collected Watson's pass just inside the Newport half, took a couple of strides forward and unleashed an unstoppable 35-yarder into the top corner.
How the Exiles must loathe the sight of the Spaniard's name on an opposition teamsheet, his last goal in a Morecambe shirt before his move in the summer of 2021 coming from the penalty spot at Wembley to end County's League Two promotion dream.
The lead didn't last long, however, with Chanka Zimba and Aaron Lewis combining in Newport's first real attack to cross low for Collins to finish confidently past Macey from just inside the area.
Mendes Gomes had been left in a heap on the far touchline in the build-up, courtesy of two dubious challenges by left-back Adam Lewis, and required treatment but managed to limp back into the action.
He didn't last long, however, with Wales Under-21 international Elliot Thorpe coming on in his place at half-time - and it didn't take long for Jerome to register the first attempt on goal of the second half, his 25-yarder whistling just the wrong side of the post 20 seconds in.
It was a full Welsh international who restored the Town's lead soon after, with Lockyer feeding Pereira on the left byline and then continuing his run into the box to take the return pass and finish emphatically into the top corner from a tight angle.
Again the visitors wouldn't lie down, and within two minutes they had hit back once more, Sam Bowen bursting through midfield and playing Zimba into the box to slam an angled drive past Macey for 2-2.
Jones made a triple substitution just after the hour, replacing Jerome, Berry and Lansbury with Carlton Morris, Harry Cornick and Gabe Osho, and great breakaway led by Cornick soon brought a chance for Thorpe, whose effort was saved well by Day.
Williams was introduced for his senior debut, in place of Burke, with 20 minutes left but it was Newport who were next to go close, James Waite finding space at the far post but unable to steer his header in from close range after a long throw was flicked on.
The midfielder didn't have to wait long for a chance to give County the lead, following up as Macey couldn't hold a shot from Zimba and slamming past the Town keeper.
The Hatters pushed for an equaliser, but save for a blocked shot from Osho in six minutes of injury-time, didn't look like forcing a penalty shoot-out and it was the side from south Wales who progressed into round two.
Goals:
LT - Mendes Gomes 30, Lockyer 50
NC - Collins 36, Zimba 52, Waite 76
Att: 3,827 (207 away)
