The Saints twice hit the crossbar in the second period and had a Steven Davis goal ruled out by VAR, but it was the home side who would prevail on spot kicks following a goalless draw in normal time.
Manolo Gabbiadini, brought on with pens in mind, missed the only effort from 12 yards as Leicester set up a last-eight tie with holders Manchester City at the King Power Stadium next month.
This fixture was originally scheduled to take place last month, but was postponed after the helicopter crash that claimed the life of Leicester owner Vichai Srivaddhanaprabha and four others.
Home boss Claude Puel made nine changes to his starting lineup, retaining just Jonny Evans and Demarai Gray from the draw with Brighton & Hove Albion, while opposite number Mark Hughes brought in six new faces on the back of his side's loss at Fulham.
The large number of changes had a clear effect on the game, as neither side created any real clear-cut openings in the first half.
Jannik Vestergaard was guilty of squandering a half-chance when heading over from a corner, while up the other end the same player diverted a Jamie Vardy cutback inches wide of his own goal at the very end of the first half.
The visitors were the side looking more likely to score, though, as Nathan Redmond struck the crossbar after taking the ball past a couple of men, before Obafemi half-volleyed the rebound over an unguarded net.
That let-off was not enough for Puel's men as Pierre-Emile Hojbjerg was next to come close with his edge-of-the-box shot that required stopping by Ward.
Gray injected some pace into the Foxes' game as the contest entered its final 10 minutes, but his direct run lacked end product as he blasted wide of the target after doing all of the hard work.
Then came the late drama - first Davis having his goal ruled out for handball against Redmond in the build-up to his tap-in, before Evans's penalty shout for a coming together in the box was also rejected by the VAR review.
There was still time for Gabbiadini to curl a left-footed free kick against the frame of the Leicester goal from the final meaningful kick of normal time, leading to a penalty shootout.
Leicester, who beat Wolverhampton Wanderers on spot kicks in the last round, again held their nerve as Mendy tucked home the winning penalty after Gabbiadini's pen was pushed around the post by Ward.
Leicester City (4-4-2): Ward; Simpson, Evans, Soyuncu, Fuchs; Diabate (Albrighton 71'), Silva (Mendy 61'), Ndidi, Gray; Iheanacho (Okazaki 61'), Vardy
SOUTHAMPTON (3-5-2): Gunn; Vestergaard, Stephens, Yoshida; Valery, Hojbjerg, Lemina, Armstrong (Davis 73'), Targett (Soares 61'); Redmond, Obafemi (Gabbiadini 91')