Chelsea needed penalties to get past Southampton at Stamford Bridge after the game ended 1-1 following 90 minutes.
Thomas Tuchel's side took the lead through a Kai Havertz header just before the break.
Reece James slotted in the decisive spot-kick as Chelsea ran out 4-3 winners on penalties after Theo Walcott and Will Smallbone failed to score for the Saints.
Arsenal eased through with second-half goals from Calum Chambers and Eddie Nketiah in a 2-0 win over struggling Leeds.
