Ronald Koeman's men fell to their fifth loss in a row at Bramall Lane, as the League One outfit progressed courtesy of Mark McNulty's second-half goal.
Southampton's misery was compounded when Florian Gardos was dismissed in the final minute for a professional foul, and captain Fonte was honest in his assessment as the south-coast club missed out on a first League Cup semi-final since 1987.
"It was not acceptable," he told the club's official YouTube channel.
"I would firstly like to apologise to the fans, they didn't deserve this. That is not us.
"Given the last games we have lost, we played and we showed that we are better than the opposition and we had the opportunities to win and deserved more from them games.
"But after Tuesday night, I am disappointed obviously and I am angry because we had a great opportunity to go and do something special and we just blew it.
"Everyone needs to take responsibility, not hide in their shell. We must stand up and be counted because there's another game in the Premier League [at home to Everton on Saturday] and we need to stop this bad run.
"The minimum we can do is try our maximum and give everything that we can, but it was disappointing because I think we should have done much better.
"I just want to apologise to the fans, because they didn't deserve it. We will be working harder to make sure this never happens again. It is an opportunity lost, but there is still a long season ahead."
