The original fixture was due to take place on Saturday, January 9, but was postponed after a number of positive COVID-19 tests at Shrewsbury.
Whichever side progresses will host holders Arsenal in the fourth round four days later.
For a while it looked as though Southampton would be handed a bye to the fourth round as a result of the coronavirus outbreak at Shrewsbury, but both clubs, the Football Association and the Premier League worked to find a solution.
Southampton head into this cup tie on the back of a 2-0 loss against Leicester City on Saturday evening, with James Maddison and Harvey Barnes netting to end the Saints' three-game unbeaten streak without conceding.
Two of those games saw Hasenhuttl's men fire blanks of their own, though, so an improvement is required in front of goal if they are to advance past Shrewsbury.
The Shrews have already had to come through a couple of rounds to make it this far, seeing off Cambridge United 2-0 and Oxford City 1-0 after extra time.
Due to the coronavirus outbreak in their camp, however, they have not played a match in any competition since December 29 - a 1-0 win over Blackpool.
Steve Cotterill's side are unbeaten in five, winning four of those matches by the same 1-0 scoreline, and are 17th in League One with games in hand to play on the teams around them.
However, Town have lost 12 of their 13 away FA Cup matches against top-flight opponents (D1) and have never won away at Southampton in nine previous attempts in all competitions (D2 L7).
Southampton, who are expected to heavily rotate on Tuesday, have also won their only two FA Cup meetings with Shrewsbury - 4-1 in the fourth round in 1952-53 and 2-0 in the first round 10 years ago.
Southampton form (all competitions): DLDDWL
Shrewsbury Town FA Cup form: WW
Shrewsbury Town form (all competitions): LWWWDW
Shrewsbury boss Cotterill is one of 20 members of the playing and coaching staff to have contracted coronavirus, meaning that he will be absent for this match at St Mary's.
The Shrews welcomed a number of those affected by COVID-19 back to training on Saturday, but it remains unclear exactly who will be available for selection on Tuesday.
As for the home side, they could be without as many as eight first-team players through injury.
Nathan Tella, who has featured as a substitute five times in the Premier League this term, is among those pushing for inclusion.
Southampton possible starting lineup:
Forster; Valery, Stephens, Salisu, Vokins; Tella, Diallo, Smallbone, Redmond; Long, N'Lundulu
Shrewsbury Town possible starting lineup:
Sarkic; Williams, Ebanks-Landell, Pierre; Daniels, Norburn, Zamburek, Vela, Daniels; Whalley, Udoh
We say: Southampton 2-0 Shrewsbury Town
Shrewsbury have not played a game in nearly three weeks and travel to St Mary's with the availability of half their squad unknown.
Southampton could have done without this game being rearranged, though it does at least provide Hasenhuttl's men with a chance to return to winning ways, which we are backing them to do with relative ease.