The visitors drew first blood in just the sixth minute of the game when Graziano Pelle picked the ball up in the middle, pushed forward through the Potters' defence and lashed a curling effort past Asmir Begovic.
Ronald Koeman's charges would go on to double their lead on the half-hour mark when a Steven Davis cross fell kindly to the unmarked Shane Long, who had the simple task of tapping into the goal to secure his first strike in a Saints jersey.
Towards the end of the first half, Stoke were able to get the ball into the back of the net, but Jonathan Walters's effort was flagged for offside.
Southampton then had a good spell around the hour mark when Long's effort following a counter-attack was thwarted by Begovic, before Victor Wanyama bore down one-on-one following some good interaction with Dusan Tadic, with the Potters stopper holding firm once again.
However, the hosts imploded when Peter Crouch was sent off for two bookings in the space of two minutes, with the subsequent free kick being duly converted by Pelle in the 88th minute to send Southampton into the quarter-finals of the competition.