The cup game had been set for January 9, but was postponed because of a coronavirus outbreak at the Sky Bet League One club.
A tweet on the official Emirates FA Cup Twitter account said: "The #EmiratesFACup third round fixture between @SouthamptonFC and @shrewsweb has been rescheduled following a decision by @FA's Professional Game Board.
"The tie will now take place at St Mary's Stadium at 8pm on Tuesday 19 January and broadcast live in the UK on @btsport."
Southampton added on their website: "This also means that Saints' scheduled visit to Leeds United for a Premier League fixture on Wednesday 20th January will be moved as a result. A new date will be confirmed in due course.
"The winner of the Shrewsbury game will advance to host Arsenal in the fourth round, with ties due to take place on the weekend of January 23rd/24th."
The FA tweeted: "We would like to thank the @premierleague, @EFL, @btsport, @SouthamptonFC, @shrewsweb and @LUFC for helping to make the rescheduling of this @EmiratesFACup tie possible."
Elsewhere, Sheffield Wednesday's next two Championship games have been suspended due to a Covid-19 outbreak.
The Owls reported a "significant number" of club personnel returned positive results for coronavirus last week.
Wednesday managed to play Exeter in the FA Cup last weekend, but will not be able to face Coventry and Wycombe, on January 16 and 19 respectively, in the league.
"The club informed the EFL that we would be unable to fulfil the fixtures following recent positive Covid-19 tests and the requirement for players and staff to self-isolate as per EFL and Government guidance in order to mitigate against the risk of further infection," a statement from the Owls read.
The news of further fixture disruption comes as Premier League referees will warn captains "the eyes of the world are on us" in a bid to stamp out goal celebrations amid tighter coronavirus restrictions.
Match officials will specifically remind Premier League players of their responsibilities through pre-match meetings with skippers, PA understands.
Chief executive Richard Masters wrote to all the Premier League clubs on Tuesday, reiterating the importance of adhering to new, stricter coronavirus protocols.
Players still celebrated goals in Tuesday night's three fixtures, but Premier League bosses are understood to expect their latest warnings to have an impact across the next round of matches.
Masters' letter would only have been received by clubs late on Tuesday afternoon, so the six teams preparing for matches might not have been brought up to speed before kick-off.
Players have been told to avoid hugging, handshakes and swapping shirts as the coronavirus protocols are raised in the wake of several outbreaks at clubs and a clutch of postponed matches.
Manchester City host Brighton and Tottenham entertain Fulham on Wednesday night, and Premier League chiefs hope their latest directives will now start to sink in.
Referees are understood to have been told by Premier League bosses to remind players that Government support for football to continue is based around compliance with the strict protocols that were enhanced still further last week.
Officials have been told to tell captains "the eyes of the world are on us and we need to set an example".
On Wednesday morning, Sports Minister Nigel Huddleston threw his support behind Masters' warning to Premier League clubs, issuing his own direct message.
"Everyone in the country has had to change the way they interact with people and ways of working," the Conservative MP for Mid Worcestershire tweeted.
"Footballers are no exception. Covid secure guidelines exist for football. Footballers must follow them and football authorities enforce them – strictly."
The Premier League's Covid-19 officers have been granted extra powers to check dressing rooms and team buses to ensure compliance with social distancing measures.