This last-16 meeting at the King Power Stadium was rearranged after the helicopter crash which killed Foxes owner Vichai Srivaddhanaprabha and four other people last month.
Srivaddhanaprabha's son, Aiyawatt, was in attendance for the rescheduled fixture and watched on as his club earned a place in the last eight thanks to a 6-5 shoot-out success.
After 90 goalless minutes and 10 successful spot-kicks, Southampton striker Manolo Gabbiadini saw his effort saved by Danny Ward and Nampalys Mendy kept his cool to send Leicester through, ensuring another difficult outing for under-pressure Saints boss Mark Hughes.
There had been drama aplenty in the closing stages of normal time, Nathan Redmond striking the crossbar for the visitors, who also saw a Steven Davis effort in the 83rd minute disallowed for handball following a VAR review.
Soon afterwards, the video replay system adjudged there had been no foul on Jonny Evans by Pierre-Emile Hojbjerg as Leicester appealed for a penalty.
But Ward proved the hero when it mattered, superbly tipping Gabbiadini's 95th-minute free-kick onto the woodwork before going on to deny the Italian in the shoot-out and help see the three-time winners through to a home quarter-final against the defending champion.