Boss Ralph Hasenhuttl targeted ending Southampton's 45-year FA Cup drought after they reached the quarter-finals.
The visitors were deserved winners at Molineux as they look to lift the trophy for the first time since 1976.
It was also the Saints' first clean sheet in five games – having conceded 16 goals in their previous four outings, including last week's 9-0 defeat at Manchester United.
Hasenhuttl said: "We are still in the Cup, which is fantastic.
"We said when we were more comfortable in the league we can focus on the Cup. We want to go for the FA Cup and this is a chance to win something.
"We haven't had an easy time in the last few weeks so it's important to win and they have been convinced about what they are doing.
"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.
"We have a south coast derby now which is always intense and we will go there trying to win."
Southampton were the better side throughout and Armstrong fired a brilliant chance wide from close range after the break before Ings opened the scoring after 49 minutes.
The striker broke through and, after John Ruddy saved his initial shot, the ball rebounded off Ings to roll into the net.
Fabio Silva shot over, Ruddy denied Nathan Redmond and Fraser Forster saved from Adama Traore before Armstrong settled the game in stoppage time.
Ruddy made a mess of Leander Dendoncker's backpass and Nathan Tella found Armstrong for the midfielder to curl the ball in off the post.
The Saints were also awarded a penalty soon after when Joao Moutinho brought down Tella but VAR overruled referee Jon Moss and gave a free-kick, with the foul occurring outside the box.
Wolves boss Nuno Espirito Santo made six changes, naming captain Conor Coady, Pedro Neto and Adama Traore on the bench. Although his selection gamble backfired, the head coach insisted he had no regrets.
"It was about the schedule, the game, what we have in front of us, solutions for the team and trying to find other options," he said.
"It's not about regrets, we decided on the moment we had and because we wanted to find other things for the team.
"Sometimes it works out, sometimes it doesn't and today it didn't. There are many things we have to decide. Everyone needs to do more, when tough moments come we need the best from everyone.
"It's a frustrating night for us, we are out of the competition because we didn't compete so well. We conceded easy goals.
"We didn't create enough and we will keep on trying and looking to improve. We wanted to go through, it was everyone's wish."