Things looked bleak for the Clarets when Grimsby substitute Danny North skipped through to put the home side ahead in the sixth minute of extra time.
But Burnley steadied their nerves and another substitute Andy Gray levelled on 108 minutes to send the tie to penalties.
And it was Burnley who showed the greater nerve from 18 yards to go through 4-2 on penalties.
The tie only came to life after a poor first half with Grimsby looking anything but a League wo side as they tore into Burnley from the Championship.
After midfielder Ciaran Toner had hit the woodwork with his cross, midfielder Danny Boshell missed a great chance of putting the Mariners ahead on 73 minutes.
Midfielder Alan Mahon went closest for Burnley with a terrific shot which was tipped over by keeper Phil Barnes but the 90 minutes ended goalless.
The excitement continued in extra time with Grimsby going ahead through North who took advantage of hesitancy to slip the ball past keeper Brian Jensen.
That could have been it but Burnley hit back with an equaliser through Gray who kept his head to finish coolly to finish from close range when Grimsby were caught out for the only time in the game.
Burnley ultimately took the spoils but the glory was with Grimsby who more than matched them and went off to an ovation from their fans.