Kepa Arrizabalaga was the Chelsea hero as he made a string of late saves to deny Saints a place in the last eight as the game at Stamford Bridge ended 1-1 after 90 minutes.
Both goals came minutes either side of half-time as Kai Havertz headed Chelsea into the lead before Southampton hit back quickly when Che Adams bundled the ball home after Kepa parried Kyle Walker-Peters’ shot.
The Spain goalkeeper saved from Will Smallbone, Mohammed Salisu and Lyanco to take the tie to a shootout.
Kepa then denied Theo Walcott from the spot and Smallbone blasted over as Chelsea progressed despite Mason Mount’s penalty being saved by Fraser Forster.
Mikel Arteta had named a strong side in the previous two rounds with the Gunners not involved in European football for the first time in 25 seasons.
But the Spaniard made nine changes from the side that started Friday’s 3-1 win over Aston Villa.
Leeds boss Marcelo Bielsa attempted to lift his side from a slump with just four changes from a 1-1 draw with Wolves on Saturday as England international Kalvin Phillips made his return from injury.