Saints looked in control at 2-0 up at half-time against the team they beat in the Premier League on Saturday, but the Potters fought back and levelled through Mame Biram Diouf with eight minutes remaining.
Stoke appeared to be the more likely winners at that stage but Peter Crouch saw red for two yellow cards in less than a minute.
The second was for a foul on Steven Davis and, from the resulting free-kick, an error by Erik Pieters allowed Pelle to grab his second in the 88th minute.
Koeman said: "We played well
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."
Pelle has been a key figure in Southampton's run of nine wins in 10 games and took his tally for the season into double figures.
"Pelle is a great player and showed his qualities, not only scoring, but as a target man," said Koeman
"Little by little the players are learning how to play with Pelle
He's been a great signing for us.
"You can't always win the football way
Stoke made it very difficult and made it a battle in second half
But my team showed spirit
Even after the disappointment of going to 2-2, the feeling was positive on the bench.
"I had the possibility to change players and I was confident if it went to extra time."
Southampton, who have never won the League Cup, will travel to Sheffield United in the last eight.
Stoke boss Mark Hughes was frustrated by the result after seeing his side make amends for a lacklustre first-half display.
"We put in a huge effort in the second half," he said
"We needed to because in the first half we didn't really get anywhere near the intensity that I was hoping for in terms of making it a real cup tie.
"The difference with the second half was marked
It was a real effort and I thought the momentum was with us.
"At 2-2 I felt we were in the ascendancy and we could go on and possibly win it from a situation where at 2-0 down you think it's going to be very hard against a very good Southampton team."
Koeman argued Charlie Adam could have been sent off at 2-1 for a challenge on Morgan Schneiderlin that earned him a booking - one of seven shown by referee Lee Mason.
Hughes felt Crouch was a little unfortunate to see two of those - with the first coming for a clash with Jose Fonte and the second the late challenge on Davis.
"I felt the first yellow card was a little bit harsh on Peter," said Hughes.
"I think he made a genuine attempt to just try to get across the lad
Whether or not the referee thought he raised his arm, I don't know.
"The second one, he's miscontrolled it and in trying to retrieve the ball he's high
When you're on a yellow card, there's always a risk you're going to get another one if you go in with a challenge like that.
"We've lost him, and that took a bit of the wind out of our sails, and then we haven't dealt with a ball into our box."
Source : PA
Source: PA