The England international raced onto a through ball from Ryan Bertrand to prod the ball past John Ruddy at the second time of asking.
However, the linesman initially ruled the goal out for offside, with Ings and his Southampton teammates dropping back into formation to await Ruddy's restart.
But the intervention from VAR proved the 28-year-old had timed his run to perfection and the goal was allowed to stand - even though you could not have told from his muted celebrations.
The Saints secured their passage into the FA Cup quarter-finals when Nathan Tella pounced on a defensive mix up between Ruddy and Leander Dendoncker and found Stuart Armstrong, who finished superbly.
Wolves lacked any sort of cutting edge and had to wait until the 74 th minute for their first attempt on target, with the loss of Raul Jimenez continuing to hamper Nuno Espirito Santo.
Saints goalkeeper Fraser Forster was, aside from one hurried clearance, a spectator, although Salisu survived a VAR review when Morgan Gibbs-White tumbled under pressure in the box.
Wolves, with just 23 goals in 23 Premier League games, struggled to pose a threat but Southampton's early dominance had waned and Saiss lifted Joao Moutinho's free kick over just before the break.
The hosts then escaped three minutes into the second half when Armstrong missed a glorious chance. The midfielder arrived unmarked in the area but managed to sidefoot Ryan Bertrand's cross wide with the goal gaping.
Ings saved his blushes a minute later when he fortuitously opened the scoring with his eighth goal of the season. The striker burst through and when his initial shot was saved by Ruddy the ball ricochet back off Ings and rolled in.
It was initially ruled offside but a VAR review showed the forward was onside.
However lucky, it was a lead Southampton deserved and the visitors now had a platform but Fabio Silva came close to an instant leveller when he fired Gibbs-White's cross over the bar.
It summed up Wolves' wastefulness and allowed the hosts to pass the three hour mark without a shot on target.
Nathan Redmond should have punished them after 67 minutes when he wasted the chance to finish the tie. The winger raced onto Armstrong's pass, only for Ruddy's smart save to turn the ball wide.
Wolves' still had hope and they pushed to send the game into extra time - with Forster saving from Adama Traore at the far post after Salisu missed Pedro Neto's cross.
But Armstrong made it 2-0 in stoppage time when Ruddy made a hash of Leander Dendoncker's backpass and Nathan Tella teed up the midfielder to curl in off the post.
Wolves then avoided further misery when VAR overturned a penalty, awarded by Jon Moss, when it showed Moutinho's foul on Tella was outside the box.
Danny Ings sent Southampton into the FA Cup quarter-finals at the expense of Wolverhampton Wanderers, but was robbed of the chance to celebrate his goal in a 2-0 win.