Southampton, who made seven changes from Saturday’s defeat at Leicester City, learned on their youth academy in the rearranged fixture to set up a tie with title holders Arsenal.
Shrewsbury Town was without boss Steve Cotterill, who is in hospital in intensive care after coronavirus swept through the League One side.
N’Lundulu grabbed the lead for Saints on 16 minutes when he took a touch around his defender before firing a low that shot that took a small deflection on its way to the far corner.
Harry Chapman, who impressed for the visitors, tested Fraser Forster just moments before Southampton took the lead, but the goalkeeper was equal to the chance from distance.
Caleb Watts was handed his own senior debut and the 19-year-old proved to be a handful for the Shrewsbury defence, linking up with James Ward-Prowse to help set up the goal.
Southampton came closest to adding a second goal after halftime, but Matija Sarkic made a point-blank save and Aaron Pierre helped the Shrews avoid disaster by clearing the ball of the line.
Second half substitute Che Adams squandered a chance to put Saints ahead 2-0, but Ward-Prowse ensured his side’s safe passage from a free-kick that beat Sarkic for pace.
Southampton (4-4-2): Forster; Vokins, Stephens, Bednarek, Valery; Watts (Jankewitz 85), Diallo (Finnigan 90+2), Ward-Prowse, Chauke (Armstrong 79); N’Lundulu, Long (Adams 79)
Unused subs: Lewis (GK), Bertrand, Walker-Peters, Ferry, Tchaptchet
Goals: N’Lundulu (16), Ward-Prowse (89)
Bookings: Chauke (46)
Shrewsbury Town (3-4-1-2): Sarkic; Pennington (Cummings 85), Pierre, Ebanks-Lawdell; Golbourne (Sears 73), Vela, Norburn (Edwards 74), Williams; Chapman (Tracey 63); Udoh (Pyke 73), Whalley
Unused subs: Burgoyne (GK), Clarke, Goss, Daniels