Grimsby have not reached that stage since 2000, while Burton last progressed beyond the third round in 2011.
Match preview
Indeed, having returned to the Football League by winning the National League play-offs in dramatic fashion last season, Grimsby fans may feel that their side are overdue an FA Cup run.
The Mariners have not gone beyond the third round of England's most famous cup competition since losing 2-0 to Bolton Wanderers in the 1999-00 season, while their best ever effort in the tournament remains losing 1-0 to Arsenal in the semi-finals in 1935-36.
Former Ipswich Town boss Paul Hurst has already cemented his legacy at the club having guided them away from the depths of non-league football, and they look set to comfortably finish in mid-table in League Two this season, with a 14 point gap to the relegation zone but eight points and nine places separating them from the play-off spots.
As such, Hurst may take Saturday's clash against Burton more seriously than perhaps expected, especially with a potentially big-money tie against Premier League opposition awaiting them in the next round.
Burton, on the other hand, are facing the daunting prospect of joining Grimsby in League Two next season for the first time since winning England's fourth tier in the 2014-15 campaign.
The Brewers only joined the Football League for the first time in their history in 2009 when winning the National League, before incredibly reaching the Championship with promotion from League One seven years later.
However, an extremely difficult start to the season left them marooned at the bottom of the table for several weeks, although improved form under their new boss Dino Maamria means that only goal difference is keeping them below the dreaded dotted lines.
Whether Maamria would view a cup run as a desirable distraction right now must certainly be up for debate, although winning can never be considered a bad habit in football, especially with considerable financial rewards up for grabs in the FA Cup for clubs such as these two.
Grimsby Town FA Cup form:
W
W
Grimsby Town form (all competitions):
W
L
W
L
L
W
Burton Albion FA Cup form:
W
W
Burton Albion form (all competitions):
D
L
L
W
W
L
Team News
After beating in-form Stockport County 1-0 last weekend, Hurst may reward the same starting XI with the chance to inflict a mini-upset on Saturday.
There is a whole week until their next league fixture against Swindon Town, so while some fringe players may be demanding an opportunity to impress him, Hurst is likely to prioritise reaching the next round.
Maamria, though, may wish to rest some key players given his side's precarious league position, especially having been thrashed 5-0 by relegation rivals Morecambe Town last weekend.
The likes of Jonny Smith, Mustapha Carayol and Joe Dodoo are all pushing for starts having come off the bench during that defeat.
Grimsby Town possible starting lineup:
Crocombe; Smith, Waterfall, Maher; Efete, Holohan, Green, Amos; Clifton, McAtee, Khan
Burton Albion possible starting lineup:
Sinisalo; Mancienne, Shaughnessy, Oshilaja; Smith, Davis, Onyango, Powell, Carayol; Dodoo, Winnall
We say: Grimsby Town 2-1 Burton Albion
With Burton unlikely to take this match too seriously considering their relegation fears, we can envisage Grimsby recording a victory to reach the fourth round of the FA Cup for the first time in 23 years.