A hugely entertaining tie ended deadlocked at 2-2 at the final whistle, with Saints progressing 4-2 on penalties after Rhian Brewster and Oli McBurnie missed from the spot in the shoot-out.
Earlier, McBurnie had levelled for United midway through the second half - with his first goal since December - after the visitors led through strikes from Ibrahima Diallo and Mohammed Salisu, who both netted for their maiden goals for the Hampshire outfit.
The game had began with an opener from an unlikely source in Enda Stevens, who put United ahead with his first goal for the club in almost 18 months in his first appearance for the club this term following double hernia surgery.
But ultimately it was Ralph Hasenhuttl's side who progressed with James Ward-Prowse, Che Adams, Diallo and Oriol Romeu netting in the shoot-out, with the latter converting the decisive penalty.
Oli Norwood and Ben Osborn found the net for United.
Reflecting on the tie, Jokanovic, who changed his entire starting 11 for the third successive match in this competition, commented: "I am satisfied with my team and they showed a great attitude.
"We played a decent game against a very complicated opponent for us and my team were fighting and created chances. They created chances as well.
"Unfortunately, we were not good with penalties, but penalties is a lottery. We are out of the competition, but in general, we must be proud.
"If you lose the game - and we lost the game on penalties - lose the game the right way. It was the right way from my team who fought. That is the sensation, even though we are out of the competition.
"A few months ago, this team were in the Premier League and we have some really good players. A lot of them have now shown they can be part of the main competition now, which is the Championship.
"We are still in the process, but there is progress, although we still need to improve."
On the performance of McBurnie, who produced a game performance full of endeavour and running, the Serb added: "Oli played a really good game. He was involved in lots of challenges and scored a goal and fought well for us.
"It was unfortunate with the penalty, but it is part of the lottery. He showed he is an important player in our squad."