In a first half of few chances, the newly-relegated hosts came closest to opening the scoring but defender Mark McChrystal failed to convert a corner at the far post.
After the break, the game opened up and the Mariners had their best chance to break the deadlock through Lenell John-Lewis but he could not find the net when the ball fell to him after an effort from Craig Disley was blocked on the line.
Town were rarely seriously tested in front of a big, 7,000-plus crowd but their 452 travelling supporters would have left satisfied with a point against one of the early promotion favourites.
Grimsby boss Paul Hurst was without six players through injury or suspension but was able to name striker Alan Connell in his starting XI, less than 24 hours after the fans’ favourite completed a short-term return to Blundell Park.
James McKeown started in goal, with Carl Magnay making his league debut at right-back and Andrew Boyce, Chris Doig and Aswad Thomas also contributing to the defence.
Jack Mackreth and Scott Brown made their first league starts for the Mariners in a midfield that also included Disley and Paddy McLaughlin, while Connell was joined in attack by John-Lewis.
Paul Bignot, Craig Clay, Ellis Humble, Caine Winfarrah and Lewis Bemrose were all named on the Grimsby bench.
The home side were playing their first game since being relegated from League Two on the final day of last season.
Former Town goalkeeper Steve Mildenhall was named in a Rovers side which also included five new signings in Stuart Sinclair, Lee Mansell, Andy Monkhouse, Daniel Leadbitter and Matty Taylor.
The hosts had the first chance of the game on four minutes when a free kick from just inside the Town half fell for striker Ryan Brunt, who drilled a low shot from the right of the penalty area that McKeown got down to stop.
It proved to be the only real opportunity at either end in a quiet opening 10 minutes, with both sides looking to get a foothold in the game.
Brunt showed good strength to collect a cross from Monkhouse and hold off Thomas before getting a shot in on 14 minutes but again, McKeown was able to make a comfortable save.
Town’s first chance saw John-Lewis flick a ball on for Connell, whose shot was blocked for the first corner of the game. The Mariners kept possession when it was cleared but Brown’s deep cross was well defended by the home side.
A home corner came on 23 minutes and they could have taken the lead from it when Mansell’s delivery went all the way through to McChrystal at the far post but his effort lacked power and McKeown blocked it with his legs.
Town had a good spell of possession around the half-hour mark which almost led to a chance for John-Lewis but the striker could not escape from Tom Lockyer inside the Rovers area.
McKeown then had to gather at the second attempt after good link-up play between Monkhouse and Sinclair led to the latter crossing from the right.
Magnay had to head a Monkhouse free kick away for a corner eight minutes before the break but Mansell’s high, looping delivery was punched clear by the Town keeper.
One added minute was played at the end of a half where there had been few genuine chances for either side.
The hosts had shaded the play and McKeown had been the busier of the two goalkeepers but the Town camp would have been mostly satisfied with how the opening 45 minutes of the season had gone.
No changes were made to either side before the restart and the Mariners had a good opportunity to open the scoring two minutes in when John-Lewis headed a free kick down for McLaughlin but his deflected volley was pushed over the bar by keeper Mildenhall.
At the other end, Thomas had to concede a corner by heading a cross from the left behind but the delivery from the flag was cleared by Disley.
Leadbitter was shown the first yellow card of the game for a foul on Thomas on the Town left on 56 minutes but the resulting free kick was poor and was dealt with by the Rovers defence.
Taylorhad a great chance to put the hosts ahead soon afterwards when he got in behind the Town defence to go through on goal but McKeown came off his line to block the shot and keep the game goalless.
John-Lewis saw a shot blocked just after the hour as the Mariners put another spell of pressure together.
And the striker should probably have put his side ahead seconds later when an effort from Disley was blocked on the line and the ball eventually fell for John-Lewis but his first-time shot from close range went wide of the near post.
Leadbitter was replaced by Jake Gosling as home boss Darrell Clarke made the first substitution, before a 20-yard drive from Sinclair flew just over the bar.
Mackreth hit a cross from the right towards John-Lewis on 70 minutes but McChrystal got there first to glance the ball away for a corner, which was dealt with by the hosts.
Ollie Clarke was the next to be introduced for Rovers, with Monkhouse making way as the game continued to open up.
Magnay did well to cut out a Sinclair cross at the expense of a corner, which was tamely headed wide by McChrystal.
The final home change saw Jamie White come on for Brunt up front as Rovers looked for something different in a bid to unlock the Town defence.
Taylorhit a 25-yard shot with nine minutes left but it was well-watched by McKeown, who got down to make the stop.
At that point, the Mariners were temporarily down to 10 men after Brown suffered a bad cut in a clash of heads but he was passed fit to continue after treatment.
Another home corner could have been costly for Town when the delivery from the flag was punched by McKeown and the ball fell for McChrystal but he could only drag a shot wide of the far post from 10 yards.
Clay replaced Connell with two minutes to go as Town looked to see out the remainder of the game and claim a point.
Bristol’s Lee Brown sent a wild volley well off-target as four added minutes were signalled by the fourth official.
The home side attacked throughout the additional period in search of a winner but Town held firm to ensure they returned to North East Lincolnshire with a share of the spoils.
The Mariners are next in action when Nuneaton Town visit Blundell Park on Tuesday.