Goals from Montel Gibson and Louis Boyd either side of half-time looked to have earned the hosts a win, but late goals from Brendan Kiernan and Kevin Lokko rescued Town a point before Lloyd Kerry's penalty miss saw Grimsby earn the extra-point on offer.
It was a night where Town pressed and probed their opponents throughout but looked to have come up short before their late, late show turned the game around in dramatic fashion.
After Saturday's victory over Tranmere in the Carabao Cup, Simon Weaver handed debuts to summer signings Lokko and Connor Kirby, who replaced Connor Hall and Josh Falkingham, whilst Kiernan and Jon Stead also came into the starting eleven.
Opponents Grimsby, who were defeated on penalties by Morecambe on Saturday, rang the changes with only Elliott Hewitt retained in the starting line-up.
It was Town, sporting their new all-blue away kit, who made the livelier start with Kiernan testing Russell in the home goal just five minutes of action. Debutant Kirby was instrumental in midfield five minutes later when he set up Ryan Fallowfield, but his pull-back flashed across the box to safety.
The home side began to force their way back into the game at the mid-point of the first half though, with Town having failed to convert their early dominance.
Will Smith was forced to clear from his own goal-line on the half-hour mark before Lokko was booked for a late challenge that brought the vocal Ian Holloway to his feet in the home dugout.
The home side's pressure paid off on 35 minutes when Tom Walker was penalised for tripping Edwards. Montel Gibson made no mistake from the spot to put Town behind.
Town responded well though with both Walker and Warren Burrell testing Russell, although both efforts resulted in routine saves.
After the break Town once again continued their trend of fast starts, with Muldoon dragging his shot wide and then Smith forcing Russell into another save.
Whilst Town continued to press their opponents, they struggled to force the home keeper into anything other than routine action. This forced Simon Weaver into a double-change with Aaron Martin and Mark Beck replacing Jack Muldoon and Jon Stead either side of the hour mark in an attempt to jump-start the attack.
This was followed by a notable home change with Louis Boyd, at just 15 years and 324 days of age, becoming Grimsby's youngest ever player when he was introduced into the action.
And it was Boyd who looked to have sealed the win for the home side when he struck a beauty from 25-yards out just minutes later to double the lead.
Town received a lifeline with ten minutes remaining when the home side were penalised for handball. Captain for the evening Burrell struck his against the post but Kiernan had the presence of mind to hit home the rebound and reduce the areas.
And barely a minute later Town snatched a point from the game when Lokko poked home following an almighty goalmouth scramble.
With the game ending 2-2 at the end of the 90 minutes, the match headed to penalties to determine which of the sides would come away with the extra point on offer. Both sides comfortably converted their efforts until Town's fifth and final effort saw Lloyd Kerry hit the post to see Grimsby win 5-4.
The result though sees Town come away with a point though and well placed ahead of their remaining two games in the competition.
Town: Cracknell, Fallowfield, Smith, Lokko, Burrell, Kiernan, Kirby, Kerry, Walker (Thomson, 75), Stead (Beck, 64), Muldoon (Martin, 58)
Subs: Falkingham, Emmett, Hall.
Goals: Kiernan (80), Lokko (85)
Booking: Lokko (34)
Grimsby: Russell, Hewitt (Jackson, jr), Ohman, Idehen, Hope (Boyd, 67), Taylor, Clifton, Spokes (Khouri, 58), Starbuck, Edwards, Gibson.
Subs: McKeown, Adlard, Grist.
Goal: Gibson (pen, 36), Boyd (74)
Red card: Khouri (90)
Report by Danny Pugsley.