Substitute Danny Crow cancelled out Ashley Barnes' opener before scoring again in the first period of extra time along with Conor Washington.
Brighton had the ball in the net inside the first three minutes but Jake Forster-Caskey was in an offside position as he headed in a cross by Kemy Agustien.
A quick break saw Newport threaten for the first time and Adam Chapman headed over at the far post with the Seagulls defence stretched.
Gaining in confidence, County forced three successive corners and much-travelled striker Chris Zebroski had a close range effort scrambled away.
But the Seagulls broke through to take the lead on 18 minutes when unselfish play from Will Buckley set up striker Ashley Barnes and he coolly rolled the ball into the net from close in.
Brighton goalkeeper Casper Ankergren came to his side's rescue by tipping over a goalbound header from Zebroski before Christian Jolley surged through and put an angled drive over.
Newport goalkeeper Jamie Stephens was forced to stretch to parry a 25-yard strike from Rohan Ince and later swooped low to hold a dangerous ball into the box by Spanish play-maker David Lopez.
Brighton continued to be threatening in attacking areas and Kazenga LuaLua saw a deflected shot go over before Buckley's low drive was taken by the scrambling Stephens.
The Seagulls should have doubled their advantage shortly before the break when Barnes put the ball into the side-netting from a tight angle after LuaLua's quick break.
Forster-Caskey then had a low shot from outside the area saved by Stephens following a cleverly worked corner from Lopez.
Barnes was off target for the Seagulls at the near post as Buckley carved out a good opportunity for the Seagulls, and then Buckley curled a shot inches wide.
Brighton continued to look dangerous and defender Adam El-Abd shot narrowly over after trying his luck from 25 yards.
The Seagulls were reduced to 10 men 23 minutes from time when Spanish defender Inigo Calderon was shown a straight red card by referee Brendan Malone following a challenge on Byron Anthony.
Both players were fully committed when challenging for a 50-50 ball and Anthony, after suffering a leg injury, had to be stretchered off the field.
Newport pulled level nine minutes from time through substitute Danny Crow, who smartly headed in a cross from Adam Chapman.
Brighton pushed hard for a winner in the closing stages of normal time and Barnes went close with a header before putting the ball over the bar after being put through by Agustien.
Striker Crow struck again to put Newport ahead in the fourth minute of extra time with a smart finish into the top corner passed keeper Ankergren after Lopez failed to close him down.
Substitute Washington pounced on a defensive mistake it make it 3-1 in the 102nd minute, coolly rounding Ankergren before putting the ball in the empty net.