Stafford, who began their FA Cup campaign in August, had only once beaten Football League opposition in 16 attempts in the FA Cup since 1975 and were playing in the first round for the first time in 20 years.
The upset was on inside three minutes when Birch's ball over the Grimsby defence found Mason's run and the firefighter drove a rising shot into the top corner.
Morgan's drive struck the base of the left-hand post and Stafford goalkeeper Russell Saunders then got down well to palm away Joe Foulkes' goal-bound strike as Grimsby hunted an equaliser.
However, the hosts were next to score when Bebeto Gomes' deflected effort was brilliantly kept out by Grimsby stopper Cartwright and Birch somehow squeezed in the rebound from an acute angle for his 12th goal of the season.
Justin Amaluzor smacked the crossbar for Grimsby after the break but Cartwright twice denied Birch from close range to prevent Stafford from disappearing out of sight.
The goalkeeper's heroics were rewarded when Rodgers rose highest to head in Morgan's free-kick and moments later Kacurri found space to nod powerfully past Saunders.
Stafford regrouped in extra time and restored their lead when Mason turned in Gomes' centre from close range, but Turi controlled a loose ball and poked in to level at the end of the first period.
Saunders was at full stretch to deny Sonny Finch after a last-gasp winner in a frantic second period and earn Stafford a shot at penalties.
But it was his opposite number, Cartwright, who would be the hero in the shootout where spot-kicks from Andy Cook, Cian Doyle and Amazulor sent Grimsby through.