The victory lifted Southampton after four successive Premier League defeats, Ings firing them ahead in the 50th minute with a freakish goal as goalkeeper John Ruddy saved the striker's close-range shot from a tight angle but the ball ricocheted off Ings and trickled over the line.
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The goal was initially ruled out for offside but awarded after a VAR check.
Armstrong sealed Southampton's win in the 90th minute with a clinical finish in off the post as he took advantage of a catalogue of errors and side-footed the ball past three defenders and the stranded Ruddy.
Southampton missed several chances to win by a bigger margin, with Armstrong shooting wide of a gaping goal from close range and Ruddy denying Nathan Redmond with a reflex save after the forward's jinking run.
Fabio Silva fired over the bar at the other end before visiting keeper Fraser Foster pulled off a superb save with his feet to deny Adama Traore, but Wolves lacked cutting edge and their late pressure proved in vain.
Next up for Saints is an away quarterfinal match against Bournemouth on the weekend of March 20th (stream all matches live on ESPN+ in the U.S.).
Saints manager Ralph Hasenhuettl said reaching the FA Cup last eight was a welcome lift for his team, who were beaten 9-0 by Manchester United in the Premier League on Feb. 2 and lost 3-2 to West Ham on the weekend.
"We haven't had an easy time in the last few weeks, so it's important to win," the Austrian told BT Sport. "In the end we deserved to win. It's not easy to create chances against Wolves, we had the better ones and we never stopped believing.
"This time we had luck with the VAR for the first time in a long time."