The Mariners struck with their three shots on target, as Harry Clifton, Danilo Orsi and Danny Amos netted before half-time to cause an upset in Lincolnshire.
Manager Rob Edwards made six changes from the side that beat Stoke on Saturday, with Elliot Thorpe making his first Luton start of the season after coming off the bench three times prior to a loan spell with Burton Albion.
Fred Onyedinma was back in the starting line-up for the first time since October after having had a big impact as a sub since his return from injury in the past fortnight.
Edwards named a midfield three of Louie Watson, Allan Campbell and Luke Berry, whilst Cauley Woodrow joined Elijah Adebayo up front, the latter who scored against Grimsby last time out.
Named amongst the substitutes was 17-year-old Axel Piesold. The former Tottenham Hotspur U18s midfielder recently signed a development contract with the Hatters after impressing on an extended trial from Dunstable Town.
Sixteen-year-old left-back Joe Johnson was also named on the bench after playing 90 minutes in the FA Youth Cup fifth round defeat to Preston at Kenilworth Road 24 hours earlier.
Adebayo saw a header drift wide of the far post from a Thorpe corner in the opening stages, but it was the hosts who took the lead in the ninth minute and it was Harry Clifton, scorer of their equaliser at Kenilworth Road last Saturday week, who smashed a left footer past Ethan Horvath after being played into the area by Danilo Orsi.
Watson picked out Adebayo with a pinpoint ball into the box on 15 minutes, but the striker's header back across goal was cleared, before Berry nodded over from a couple of yards out five minutes later after Osho had headed another Thorpe corner goalward.
Onyedinma drove into the box from the left flank in the 25th minute, only to see his cross-shot parried by Max Crocombe in the Grimsby goal.
But by the 27th minute it was 2-0 to the hosts, with Orsi collecting the ball from a lay-off from a throw and slamming another angled shot into the far corner of Horvath's net.
Town mounted an immediate response, Thorpe whipping in a dangerous cross from the right before Bell and Onyedinma did likewise on the left, but each time the Mariners' backline stood strong.
Grimsby midfielder Bryn Morris was first into referee Sam Barrott's book for tripping Thorpe in full flight just outside the box, but Watson's curling free-kick was headed clear.
Town's first real chance came in the 39th minute when Osho made progress down the right and curled in an inviting cross that Adebayo met with a firm downward header, only for Crocombe to recover from a slip to save with his feet.
Watson and Grimsby left-back Amos were the next to see yellow for over-zealous challenges, then the hosts fashioned another shooting chance for right wing-back Michee Efete in time added on, but Onyedinma got back to deflect the ball behind for a corner.
Four minutes into that added period, the Mariners made it three with Amos' inswinging free-kick from the right beating everyone and going in off the far post to complete a horror half for the Hatters.
Edwards made two changes at the break, introducing Alfie Doughty and Pelly-Ruddock Mpanzu for Thorpe and Watson, and it was the former who created the first two chances of the second half, first crossing for Woodrow to flick a header goalward that Crocombe did brilliantly to save, then seconds later for Adebayo to head straight at the home keeper.
Two more changes followed just before the hour, with Jordan Clark on in midfield in place of Campbell, with a third striker thrown into the mix as top scorer Carlton Morris replaced left-sided centre-back Bell, with Woodrow dropping into the hole.
In the 61st minute the changes almost paid off, as Doughty picked out Morris with a diagonal ball and the striker laid it back for Berry to strike a 25-yarder that Crocombe did well to hold onto.
Doughty curled over himself on 63 minutes, then Woodrow crossed for Morris to send a header on goal two minutes later, Crocombe gathering comfortably.
Town were by now camped in the hosts' half, Clark whistling a shot past the post, although Grimsby posted another reminder of the threat they had carried throughout the two ties when Morris fired inches wide, just before Woodrow volleyed over the bar at the other end with quarter-of-an-hour to go after being teed up by Adebayo just outside the box.
Mpanzu came as close as anyone to getting a goal back when he sidefooted just past the post after a clever one-two with Woodrow on 88 minutes, but ultimately it was a night to forget for the Hatters, and quickly turn the focus back to the Championship and Saturday's trip to Coventry.
Goals:
GT - Clifton 9, Orsi 27, Amos 45+4
Att: 7,051 (466 Hatters)