Southampton took the lead at Goodison Park through Kyle Clinton before Michael Donohue’s late effort forced the tie into extra-time.
While the two sides couldn’t be separated over 120 minutes, it was Southampton who prevailed on penalties, converting five of their spot-kicks to Everton’s four.
The young Foxes had already booked their spot in the next round at King Power Stadium on Monday evening.
A date and time for the clash will be confirmed on LCFC.com in due course.