After a tight game, an Ismail Yakubu header two minutes from full-time proved the difference between the two sides on the night and means Newport lead 1-0 at the halfway point in the tie.
The men from south Wales had been the better side in the first half and had looked the more likely to open the scoring.
But Grimsby regrouped, came out stronger after the break and could have had the advantage when captain Craig Disley hit the post with a header.
The visitors had not looked as threatening in the second half but they took their chance from a late free kick, which means the Mariners must win the second leg at Rodney Parade to have any chance of progressing to the final.
Town bosses Rob Scott and Paul Hurst made three changes from the side that beat Newport in the final match of the regular league season on Saturday.
James McKeown, Shaun Pearson and Andy Cook returned to the starting XI, with Greg Fleming, Andi Thanoj and Frankie Artus making way.
McKeown started in goal, with Sam Hatton, Pearson, Tom Naylor and Aswad Thomas forming the defence.
Cleveland Taylor began on the right side of midfield, with Tom Naylor moving forward to partner Disley in the middle and Jamie Devitt starting on the left.
Cook was chosen alongside Liam Hearn in attack as the Mariners started with their familiar 4-4-2 formation.
Thanoj was named on the Grimsby bench, along with Bradley Wood, Dayle Southwell, Ross Hannah and Marcus Marshall.
Newport’s starting line-up included just four of the team that began the match at Blundell park on Saturday as manager Justin Edinburgh brought most of his star names back in.
Grimsby-born Danny Crow led the Exiles’ attack, with Christian Jolley also likely to be a threat in a front three for the Welsh side.
Town kicked off attacking the Pontoon Stand end of the ground and they had the first chance to get the ball into the box when Naylor was fouled on halfway on three minutes. Hatton’s free kick was met by Pearson but his header was stopped by visiting keeper Lenny Pidgeley.
Newport’s first effort on goal came when McKeown could not prevent a deflected shot by Jolley from going out for a corner. Michael Flynn’s left-wing delivery was headed clear by Disley.
A foul by Hatton on Alex Gilbey gave the away side a free kick midway inside the Town half on seven minutes but Flynn’s ball towards the far post beat everybody and went out for a goal kick.
Former Wimbledon striker Jolley won another corner soon afterwards and from it, the visitors could have taken the lead but defender Ismail Yakubu sent a free header off-target at the far post.
The Mariners had a chance of their own on 12 minutes, when a Hatton free kick was not cleared and the ball was worked back into the box, although Disley’s shot lacked the power needed to beat Pidgeley, who dived to stop it.
A snap-shot by full-back David Pipe flew over McKeown’s bar as Newport continued to look the brighter of the two sides in the early stages.
At the other end, Grimsby’s best opportunities were coming from free kicks and another Hatton set-piece from halfway resulted in Hearn shooting wide.
Pipe sent a fierce cross-shot just wide of the mark on 21 minutes as the visitors continued to look dangerous in attack, before Disley was shown the first yellow card of the game for a trip on Jolley.
Lee Minshull picked up Newport’s first booking soon afterwards for a foul near the halfway line and another foul by the County midfielder gave Town a good chance to get the ball into a dangerous area, although Hatton’s free kick was headed away.
The Mariners kept up the pressure though and they had another free kick when Devitt was fouled on the left on 27 minutes. The Irishman drove the ball in towards the far post and keeper Pidgeley had to tip it over his bar for a corner.
Jolley won a corner for Newport down the right on the half-hour but Andy Sandell’s ball was headed over the bar by Yakubu.
Another away corner on the right came five minutes later but the ball from the flag was cleared at the near post.
A run from Devitt down the left brought the home supporters to life but the Hull City loanee could not make it count, losing the ball after finding his way into the penalty area.
Crow and Gilbey combined to create another opening for the away side but Gilbey looked to find Jolley instead of pulling the trigger and the chance was gone.
One added minute was signalled to be played at the end of the half, with Town on the attack as Devitt and Thomas both had shots blocked.
But it was Newport who had looked the side more likely to open the scoring over the first half as a whole, meaning the Town management team would probably have been happy to go in level at the interval.
No changes were made before the restart and Town, playing with the wind at their backs, had the first opportunity of the second half with an early corner, which led top another one after good work by Hearn.
Taylor was booked for a handball as Newport looked to launch an attack but the Mariners had started the half well and a half-chance came and went when Devitt dragged a shot wide from 25 yards three minutes in.
At the other end, a good challenge from Hatton denied Crow after a through-ball by Gilbey. The resulting corner was headed away by Miller.
A low cross from Thomas on the left was cleared, before McKeown came well out of his penalty area to eventually clear after initially losing out to Crow.
The first substitution of the game was made by the Mariners on 57 minutes and saw Thanoj come on in place of Devitt.
The sub’s first action of note was to send a 25-yard shot well over Pidgeley’s bar just before the hour.
Town won a corner on the left on 63 minutes but the kick from Thanoj was headed away by Pipe.
Grimsby had improved after the break and were now the side asking the questions of the opposition defence, although keeper Pidgeley was still yet to be called into serious action.
Conor Washington replaced Crow in the Newport attack with 25 minutes left on the clock.
Another Town chance came when Thomas got the better of Washington on the left and pulled the ball back for Taylor on the penalty spot but the winger scooped his shot a long way over the bar.
Marshall replaced Taylor on the Town right on 72 minutes and his first action was to send over a right-wing cross that was headed clear by Minshull.
Grimsby were inches away fro taking the lead when the ball bounced around in the Newport box after a chipped cross from Thomas on the left and a header from Disley hit the post before it was cleared by defender Tony James.
Town were on the attack again on 79 minutes as Hearn found Naylor on the left and he cut in on his right foot but sent a cross out of play at the far post.
Jolley, who had been much quieter after the break, hit a cross-shot wide of McKeown’s goal soon afterwards as both sides looked for what could prove to be a vital goal.
Another chance for Jolley came when he picked up a long through-ball from Sandell and cut inside but shot wide of the far post.
Good work by Hearn then saw Town win a corner with seven minutes left but Thanoj could not beat the first defender with his right-wing delivery.
Scott Donnelly came on for Gilbey as the visitors made another change on 85 minutes, before Cook got on the end of a deep cross from Hatton but could only direct his header into Pidgeley’s arms.
Robbie Willmott replaced Jolley with three minutes to go as the away side prepared to take a free kick on the right, close to the Town penalty area. From it, Sandell delivered a perfect ball and YAKUBU rose at the far post to open the scoring.
Five additional minutes were signalled by the fourth official and Town pushed forward in search of an equaliser, with a Hatton free kick cleared for a corner.
A cross from Thomas on the left was sliced away for another corner but it was well defended by goalscorer Yakubu, before a shot from Miller lacked power or direction and trickled wide.
Town kept pressing but another Thomas cross was headed clear and a poor back-pass from Pearson conceded ground and gave Newport a throw in an attacking area.
That proved to be the last opportunity of the game, meaning Newport hold the advantage after the first 90 minutes.
Grimsby are certainly not out of the tie though and they will still remain confident of turning things around when they make the long journey to Wales for the second leg on Sunday.