The Saints raced into a two-goal lead at the Britannia Stadium, before Steven N'Zonzi found the target four minutes after the interval to change the complexion of the game.
Stoke eventually levelled up through Mame Biram Diouf, although the hosts imploded late on when Peter Crouch was dismissed, with Graziano Pelle bagging his second goal moments later to snatch the victory.
"We played well," Koeman is quoted as saying by BT Sport. "It was more difficult when they made it 1-2 after half time. In the first half we scored two great goals and were much the better team on the pitch.
"They showed more character after half time, real fight, but we created three big chances for the third and didn't take them. They were physical and brought Crouch in, and it's difficult to defend. It's not the way I like to play but that's Stoke.
"Maybe after the red card for Crouch we were a little lucky but we deserved to win, we had more chances than Stoke."
Southampton will now take on Tottenham Hotspur in the quarter-finals.