The home side started the better of the two in front of the live television cameras and the Mariners had Shaun Pearson to thank for a superb block which kept the scores level at the interval.
After the break, the visitors grew in confidence and the game was much more even but ultimately, neither side could force a breakthrough.
New loan signing Alan Goodall came in for his Town debut at left-back in place of the suspended Aswad Thomas.
His inclusion was the one of two changes to the starting XI that lost at home to Nuneaton Town last time out, with Scott Kerr replacing AndI Thanoj in the other alteration.
James McKeown started in goal, with Paul Bignot, Pearson, Chris Doig and Goodall forming the defence.
Craig Disley, Kerr and Patrick McLaughlin were the midfielders, with Alex Rodman and Joe Colbeck lining up on either side of Lenell John-Lewis in attack.
There was no place on the Grimsby bench for Liam Hearn, although Ross Hannah did return, alongside Andy Cook, Clayton McDonald, Scott Neilson and Thanoj.
Luton started brightly and they won the first corner in the third minute, although centre-back Steve McNulty headed wide from it.
The home side were keen to attack and a cross from Ronnie Henry on the right eluded striker Paul Benson two minutes later.
Andre Gray won another corner on the right but it was dealt with by Doig, before Town’s first attack of note saw Rodman win a corner after a good run by John-Lewis.
Colbeck was shown the first yellow card of the game for a foul on Jake Howells as the hosts continued to make the early running and Benson almost put them ahead from a corner but the forward headed wide.
Winger Jonathan Smith cut inside Bignot to fire a shot from 25 yards on 17 minutes but McKeown got down to make a comfortable save from it.
Benson was the next to go close, although he missed the target with a snap-shot after Doig tried to head the ball clear as the home pressure continued.
A foul on John-Lewis inside the Luton half gave the Mariners a chance to get the ball into the danger zone but Doig’s header from Bignot’s free kick went through to home keeper Mark Tyler.
Another foul on the striker by defender Alex Lacey gave Town another set-piece opportunity, this time in a central position, but Rodman’s 35-yard shot cleared the crossbar.
At the other end, midfielder Luke Guttridge got on the end of a Henry cross from the right but his header flashed wide of McKeown’s post.
Disley dragged a low shot wide from long range for Town eight minutes before the interval, before good defending by Bignot denied Howells at the expense of a Luton corner.
The Mariners had shown little in terms of attacking threat over the course of the half but they had done well to contain their hosts after an early spell of pressure.
McLaughlin almost found an opening for a shot on the edge of the Luton box as the game moved into two minutes of stoppage time but Lacey did well to challenge the former York midfielder.
Then, Town had Pearson to thank after Benson appeared to have put the home side ahead by slotting past McKeown, only for the defender to get back and make a block to keep the game goalless at the break.
Lacey was replaced by James Stevenson for Luton before the restart and Town’s first chance of the second half came six minutes in, when Rodman headed a looping cross from the right wide.
Stevenson fired wide from long range soon afterwards, before Pearson was booked for dissent.
The game had been much more even since half-time as the hour mark passed, with both teams struggling to create clear-cut chances and defences on top.
Rodman and Kerr created an opportunity for the Mariners on 65 minutes but Kerr’s shot from 20 yards was blocked by Andy Parry.
Howells forced a corner at the other end two minutes later but Luton could not create anything from it, as the delivery was dealt with before Colbeck was fouled.
Gray and Guttridge combined for the hosts to create another shooting opportunity but Guttridge was well off-target from 25 yards.
That proved to be Gray’s last action of the evening as he was replaced by former Southport danger man Shaun Whalley for the last 20 minutes.
Stevenson went close for Luton on 73 minutes when he shot from 20 yards but could only watch the ball drift slightly wide of McKeown’s goal in what was possibly the best chance of the half so far.
Town had a corner with 14 minutes to go but McLaughlin’s delivery was cleared and the deadlock remained in place.
Howells missed the target with a curling shot from the edge of the penalty area, before Grimsby’s first substitution saw Neilson replace Rodman on the left.
The Mariners were growing in confidence as the clock ticked down and Bignot thought he should have had a penalty when he appeared to be hauled down inside the Luton box, only for the referee to instead give a corner.
Alex Wall replaced Guttridge in what was the home side’s last change, with boss John Still changing to a 4-4-2 formation for the last six minutes.
The substitute’s first touch forced McKeown into a save at his near post but the Town stopper was equal to it.
Goodall fizzed a low ball into the box on 87 minutes in the hope it might lead to a goal-bound touch but McLaughlin could not direct it on target.
A double change for Town saw Cook and Thanoj come on for Colbeck and Disley, who picked up an injury.
Three added minutes gave both teams an extra opportunity to find a winning goal.
But try as they might to find a way through, both defences stood firm and it was a point apiece.
Town are next in action when they face Braintree at Blundell Park on Saturday.