The win sets up a last eight away tie against Championship side Bournemouth on the weekend of March 20.
Ings needed a bit of luck to score his first goal since January 4 when he broke the deadlock in the 50th minute with a bizarre goal as John Ruddy made the initial save but the rebound came back off the striker and rolled into the net.
Initially ruled out for offside, Southampton’s VAR fortunes changed as the goal was awarded upon a check.
Armstrong then sealed Southampton’s passage in the competition in the 90th minute, turning in a sensational low cross from second-half substitute Nathan Tella with a side-foot to make it 2-0.
Wolves managed just one shot on target in the match, which was dealt with by goalkeeper Fraser Forster, denying Adama Traore.
Ralph Hasenhuttl handed a long awaited debut to Mohammed Salisu six months after the defender arrived at the South Coast club.
Southampton, who entered having suffered four consecutive defeats in the Premier League, was boosted by the return of full-back Kyle Walker-Peters ahead of a a match up against Wolves in the league.
Wolverhampton (3-4-3): Ruddy; Kilman, Saiss, Dendoncker; Jonny (Ait-Nouri 46), Neves, Moutinho, Hoever; Vitinha (Neto 69), Fabio Silva (Willian Jose 64), Gibbs-White (Traore 63)
Unused subs: Rui Patricio (GK), Coady, Semedo, Cundle, Otasowie
Bookings: Gibbs-White (59), Traore (90+2)
Southampton: Forster; Bertrand, Salisu, Bednarek, Walker-Peters; Romeu, Ward-Prowse; Armstrong, Djenepo (Tella 87); Ings (Adams 83), Redmond
Unused subs: Lewis (GK), Vestergaard, Stephens, Finnigan, Chauke, Watts, N’Lundulu
Goals: Ings (49), Armstrong (90)
Bookings: Walker-Peters (52), Romeu (88)