A Mariners side short of attacking options started brightly but fell behind to a Paul Marshall free kick on four minutes.
After that, Town grew into the game and they drew level through Neilson just before the quarter-hour.
Grimsby had chances to take the lead but also had to soak up spells of Halifax pressure in an open game in West Yorkshire.
But they managed to stop the rot after two consecutive losses and would have been content with a point against a team expected to be challenging towards the top of the table.
Town boss Paul Hurst was forced to make changes after losing both of his starting strikers to injury in Saturday’s home defeat by Torquay United.
That meant Ross Hannah started the game, with Jack Mackreth also coming into the XI with Jon-Paul Pittman and Lenell John-Lewis both ruled out.
They accounted for two of the four alterations to the Grimsby side, with Chris Doig and Craig Clay also coming in as Paul Bignot and Patrick McLaughlin dropping to the bench.
James McKeown started in goal, with Aristote Nsiala moving to the right of a back four that also included Shaun Person, Doig and Carl Magnay.
The midfielders were Mackreth, Clay, Craig Disley, Scott Brown and Neilson, who would also look to support lone striker Hannah.
Joining Bignot and McLaughlin on the substitutes’ list were Ellis Humble, Caine Winfarrah and Paul Walker.
The Mariners started well and they won an early corner when an effort by Pearson was deflected over the bar after a free kick from Neilson was headed on by Nsiala.
But their positive opening was not to last long, as Halifax opened the scoring in the fourth minute. A poor back-pass from Nsiala was not fully cleared by McKeown and after Doig conceded a free kick, MARSHALL stepped up to fire it into the net.
Midfielder Lois Maynard missed the target with a header, before another home attack was stopped when striker Scott Boden was caught offside.
A Town corner on 11 minutes was cleared as the visitors looked to establish themselves in the game.
And they had third equaliser three minutes later, when a pass from Nsiala found NEILSON and he worked an opening on the edge of the box before smashing a shot that beat home keeper Matt Glennon and went in off the post.
Another run through the middle by Neilson resulted in a second chance for the attacker on 17 minutes but this time, his shot went the wrong side of the post.
Halifax boss Neil Aspin was forced into making an early substitution soon afterwards, with Kevin Roberts replacing the injured James Bolton.
Mackreth was blocked after a good pass from Clay as Town started to get on top in the game and after two throw-ins in quick succession, a shot from Neilson missed the target.
Roberts then got to the ball ahead of Hannah as the striker looked to get onto a through-ball and put himself one-on-one with Glennon.
A foul by defender Matty Pearson on Neilson gave Grimsby a free kick on 28 minutes but the offside flag went up as Nsiala headed wide from the delivery.
At the other end, a free kick sent into the box by goalscorer Marshall was dealt with by McKeown, before a pass from Brown found Clay but the former Chesterfield midfielder could not control it and was flagged offside in any case.
A good passage of play from the Mariners just after the half-hour led to Hannah letting fly with a long-range shot that flew narrowly wide of the goal.
Quick thinking by Neilson from a free kick led to a Town corner and before it could be taken, Brown and Halifax’s Marc Roberts were both booked. The flag-kick was cleared.
Neilson was proving to be the main threat for Town, taking advantage of plenty of space being afforded by the hosts in midfield.
A Halifax long throw four minutes before the break was half-cleared by Magnay as far as Boden but he could only volley over the bar at the far post.
A second home substitution came a minute later, with Danny Schofield replacing Maynard, who was forced off the pitch with an injury.
Town had the ball in the net on 45 minutes when Neilson held onto the ball before playing it to Disley. The skipper’s shot was saved by Glennon but when Brown converted the rebound, the offside flag went up against him and the goal was chalked off.
Three added minutes were played at the end of the half and Halifax won a corner off Pearson, which led to McKeown having to get fingertips to a header from Steve Williams to tip it onto the bar and away for another corner. This time, Disley managed to block Richard Peniket’s header.
That proved to be the last chance of a first half which Town had grown into after falling behind early. The Mariners had got themselves level and had created the better chances to get the next goal.
No changes were made before the restart and an early Halifax free kick near the halfway line ultimately did not cause any problems as Adam Smith shot wide from outside the box.
A spell of possession from the home side on 50 minutes saw them create a chance for Smith but his chipped shot was comfortable for McKeown.
Grimsby’s first chance of the half saw Neilson combine with Hannah, who tried to lift the ball over the advancing Glennon but failed to get the height needed to beat the keeper.
Pearson was booked for a foul on 55 minutes but the resulting free kick led to a foul on Doig, which brought the danger to an end.
Doig then did enough to put off an attacker before Clay cleared the ball as Halifax launched another attack.
Boden then headed wide as the home pressure began to build, before the striker saw a shot comfortably saved by McKeown.
At the other end, Magnay hit a cross-shot after a move involving Hannah and Neilson but the former Gateshead defender could not hit the target.
Disley had to concede a corner on 69 minutes but Williams headed wide from it under pressure from Nsiala.
Town were still looking to attack and a cross from Magnay was caught by Glennon soon afterwards, before a free kick at the other end led to Roberts heading over the bar.
Nsiala was booked for kicking the ball away before a Halifax throw with 20 minutes to go.
Neilson was fouled to give Town a free kick three minutes later but his ball into the box was taken by Glennon.
A shot from Mackreth was blocked, before the former Macclesfield winger had a good chance to make it 2-1 after good work by Neilson but he could only shoot at the keeper.
The Mariners were having a good spell and they won a corner on 78 minutes, which led to a Pearson header being cleared off the line. The ball was worked back to Brown but this time, his cross was cleared.
A foul on Brown three minutes later gave the visitors a free kick in a promising position on the right but Neilson’s floated ball was headed away. Town retained possession and eventually, a cross from Neilson was headed onto the roof of the net by Disley, who was in an offside position.
A late challenge by Mackreth gave Halifax an opportunity to get the ball into the danger zone with seven minutes to go but the free kick was dealt with by the Grimsby defence.
The final home substitution saw Smith replaced by Ross Dyer, with the men in blue doing most of the attacking as the clock ticked down.
A cross from Roberts was pushed away for an unnecessary corner by McKeown but the Town keeper made amends by saving the header which came in from the flag-kick.
The Mariners were also still pushing for a winner and a ball from Clay sent Hannah through, although he lacked support. Town kept possession though and a header across from Magnay as just behind Brown.
Three minutes of added time were signalled and the home side won a free kick for a foul by Pearson in the second of those minutes. Marshall’s ball was headed into the side netting by Williams.
McLaughlin replaced Hannah for Town and another Halifax chance came when a Matty Pearson overhead kick dropped just over the bar.
That proved to be the last chance of the game, leaving the Mariners to claim a precious away point.
Town are on the road again on Saturday, when they face Kidderminster Harriers.
Highlights on Mariners Player soon.