Liverpool - Southampton beat Everton 4-3 on penalties to move into the League Cup fourth round after a 1-1 draw at Goodison Park on Tuesday.
Saints defender Cedric Soares scored the decisive spot-kick after misses from Everton's Richarlison and Theo Walcott.
Walcott's 85th-minute equaliser had sent the tie to a shoot-out after Danny Ings put Southampton ahead just before half-time.
Mark Hughes' side, who have won just once in the Premier League so far this term, will travel to Leicester City in the last 16.
It means Everton have now been eliminated in six of their last seven League Cup ties against top-flight opposition.
Marco Silva's side were nearly caught out in the third minute when Ings latched onto a simple long ball, but Everton goalkeeper Maarten Stekelenburg rushed from his line to save with his legs.
Everton's Bernard was making his first start and the Brazilian forward's low drive forced Saints keeper Angus Dunn into a diving save.
Southampton went ahead in the 44th minute as Nathan Redmond's diagonal ball in the area was collected by Ings, who guided his shot inside the far post.
It was the 26-year-old's second goal in consecutive appearances at Goodison, after the on-loan Liverpool striker netted in Southampton's 2-1 defeat in August.
The introduction of Brazilian forward Richarlison inspired Everton and fellow substitute Walcott gave them a lifeline, collecting Oumar Niasse's pass to equalise.
But in the shoot-out, Richarlison was first to miss, blazing over after a stuttering run-up.
Stekelenburg denied Matt Targett, but Walcott could not beat Gunn and Cedric duly slotted in the winner.