The Mariners dominated the game against their part-time opponents and created several chances but could not make any of them count.
Town had almost all of the possession throughout as Altrincham looked to defend and add to their impressive weekend win at Halifax with another positive result.
In front of a sparse crowd, the hosts kept pushing forward to force the elusive opening goal but ultimately, it never came and they were forced to settle for a point.
Three changes were made to the Town XI after Saturday’s 1-1 draw at Dartford, with Aswad Thomas returning from injury as Craig Clay and Scott Neilson were both restored to the starting line-up.
They replaced Ross Hannah, Craig Disley and the suspended Toto Nsiala, who picked up his fifth booking of the season in Kent.
James McKeown started in goal, with Carl Magnay, Shaun Pearson, Danny Parslow and Thomas forming the defence.
Jack Mackreth, Clay, Scott Brown and Patrick McLaughlin were the midfielders, with Neilson supporting Lenell John-Lewis in attack.
Hannah was selected on the bench, along with Paul Bignot, Rhys Oates, Paul Walker and Caine Winfarrah.
Altrincham made the trip from Greater Manchester looking to make it two away wins on the bounce after their impressive 3-1 success at Halifax last time out.
They would look to prolific striker Damian Reeves to add to his seven goals so far this campaign, while further firepower could come from the bench in the shape of Kyle Perry.
Town began the game in attacking mode, with Thomas finding space down the left to get an early cross in and McLaughlin sending a low shot just wide from 20 yards.
Brown tried his luck with a speculative shot from 30 yards on four minutes as Grimsby’s quick start continued but he could not hit the target and the ball flew wide.
Another Thomas cross brought the game’s first corner three minutes later and after McLaughlin’s low ball was eventually cleared back to him, he drove a cross over which was dealt with by defender Gianluca Havern.
Brown flashed a shot just wide of visiting keeper Stuart Coburn’s goal, before Altrincham’s first attempt at goal saw Greg Wilkinson fire an effort from 40 yards which did not trouble McKeown.
A half-chance for the Mariners came when Neilson’s cross found John-Lewis on 13 minutes but the striker could only help the ball on to Coburn after it came to him quickly.
At the other end, an excellent cross from Shaun Densmore on the right was met by Simon Richman but his header went over the bar.
Good play by Brown saw him win the ball inside the Altrincham half and play it through for John-Lewis, although the move ended when the offside flag went up against the striker.
A cross-shot by Richman was put behind for a corner on the visitors’ left but it did not lead to any danger in front of goal as Brown hacked the ball clear.
Mackreth missed the target with a 20-yard shot after a move which was started by Neilson and also included McLaughlin as the hosts pressed for an opening goal.
John-Lewis almost got that goal after a run through the middle which saw him get through several defenders before curling a shot that dropped narrowly wide.
A foul on John-Lewis gave Grimsby a free kick on the corner of the Altrincham penalty area but Neilson’s delivery was headed away by Havern.
Mackreth’s low cross from the right on 30 minutes found John-Lewis but he could not control it and the ball went through to Coburn.
Town had spent almost all of the opening half-hour attacking and had put together some good passages of play but they would have been disappointed not to have worked the keeper more.
Coburn was finally called into serious action two minutes later when a low cross from Mackreth was not dealt with and Neilson hit a shot that forced the goalkeeper to dive and tip the ball away for a corner.
The away side looked to have created a rare chance on 37 minutes after Clay lost possession in midfield but the offside flag went up against Reeves as he looked to get onto a through-ball.
Brown’s delivery found Thomas but he could not direct his header towards goal. Reeves was shown the first yellow card for delaying the taking of a Town free kick, before a Neilson drive from inside the box was blocked for a home corner on the left.
A foul by Pearson on Reeves gave Altrincham the chance to get the ball into the danger zone but Nicky Clee’s cross was blocked by Magnay after Densmore’s initial free kick was headed out to the left.
One added minute was played at the end of the half and Reeves hit a shot harmlessly wide from 25 yards as the visitors had the last chance of the opening period.
That meant it remained goalless at the interval, with Altrincham perhaps the happier of the two sides to be going into the dressing rooms level.
Town had dominated possession and territory but had not caused enough problems for Coburn in the away goal.
Peter Cavanagh replace Sean Williams for the visitors before the restart and an early Brown free kick from the right was punched clear by Coburn.
Midfielder Jake Moult then tried to chip McKeown from long range but could only watch the ball drift over the bar.
That turned into a second corner on the right but this time, the keeper collected it at his far post. Coburn had to push a long-range shot from Magnay away for a corner as it swerved in front of him.
A foul by Magnay on Clee gave the away side a free kick inside the Town half but Adam Griffin’s ball was cleared by Pearson.
A corner came soon afterwards but it was easy for McKeown, who caught the ball at his far post.
Wilkinson hit a 30-yard shot straight at McKeown, before the Mariners could have taken the lead when a cross from the right fell for McLaughlin but his header missed the target.
Thomas forced a good save out of Coburn after a through-ball from Clay but the resulting corner from Brown was poor and Altrincham cleared.
John-Lewis dragged a first-time shot wide after a long pass from Neilson, after two dangerous-looking crosses into the Altrincham box had come to nothing.
Perry was sent on for the visitors in place of Wilkinson on the hour, just after they had shown some signs of promise on the counter-attack.
Magnay’s shot hit the wall. Cavanagh was booked for a foul on Clay in midfield and the free kick soon turned into another one in a position closer to goal after a foul on Neilson.
Town looked to create a chance when McLaughlin passed to Mackreth on the right on 66 minutes but his low cross was too far away from John-Lewis for the frontman to make contact.
Mackreth was having plenty of joy down the right but his crosses were not being attacked, leading to frustration on the Town bench and in the stands.
A shot from Thomas was deflected away for a Grimsby corner but Pearson’s header from it was cleared.
John-Lewis and McLaughlin were replaced by Hannah and Oates as Town boss Paul Hurst made a double substitution with a little less than 20 minutes to go.
Ryan Crowther then came on for Clee for Altrincham with 14 minutes left on the clock.
Brown almost got on the end of a through-ball from Thomas inside the away penalty area soon afterwards but Coburn was alert to the danger and raced off his line to gather.
Hannah hit a shot wide from 18 yards as Town kept searching for that elusive opening goal, with Altrincham now reduced to defending deep and clearing their lines.
A free kick from Neilson midway inside the visitors’ half was headed on but only through to Coburn.
A rare chance for the away side to win the game then came on 84 minutes when Perry’s cross found Reeves but he could not capitalise, sending a free header to McKeown.
At the other end, a drilled shot from Magnay was well-stopped by Coburn after a driving run from Brown through the middle.
Brown hooked the ball back into the box but Coburn collected it. Mackreth won a corner with a minute to go but Pearson’s header from it was cleared.
Two added minutes were signalled by the fourth official and in the first of them, the away side won a free kick when Crowther was brought down by Mackreth.
Town kept pushing for a winner but it was not to come and the referee’s final whistle meant the Mariners were left with just one point and a third game without a win.
They will look to break that sequence when they travel to Wrexham on Saturday.