The Portuguese defender's headers book-ended an own goal by defensive partner Jos Hooiveld and a late effort from substitute Kieran Richardson at St Mary's.
Fonte's goal at the death was enough to secure a point for Saints, but his overriding feeling was one of frustration after they let slip a comfortable-looking lead to draw 2-2.
Nigel Adkins' side have now shipped 20 goals - a Premier League high - and centre-back Fonte has had enough.
"I am obviously very happy to score my first Premier League goals, but it is mixed emotions because I prefer to keep a clean sheet," he said.
"I am a defender and every defender is very disappointed with this.
"I know it is not only the defenders that are to blame, but as a team have to defend better and stop this.
"Other games, not just against Fulham, we have been comfortable and then there is a period in the game that we concede one or two goals.
"That has to stop and we will work very hard on the training ground to once and for all start getting some clean sheets."
Such inability to thwart opponents has seen Saints boss Adkins rejig his back five in recent weeks and bring in new faces.
Many of those defensive options played in the second tier last season and Fonte has backed their ability to make the step up.
"No-one is more disappointed than us, don't get me wrong," he said.
"We are very upset with it because last year we had one of the best defences in the league.
"I know it is the Premier League but if you look at the goals we are conceding it does not matter if it is the Premier League or Championship. It is basic mistakes.
"That is something we were not doing last year so I think we can improve massively.
"I think we can keep clean sheets from now on. No doubt about it.
"It's just a question of concentration and keep working very hard on the training ground."
Fonte was left "disappointed" after a match he felt Southampton should have won - a feeling shared by Fulham left-back John Arne Riise.
"We were poor in the first half, we didn't control the game at all," the Norwegian said.
"In the second half we went 2-1 up and I was sure we were going to win the game and deserved it in the end.
"But playing Southampton, we knew they are good at set pieces and they scored from them today [Sunday]."
Riise played an important role in the comeback as it was his fierce drive that went in off Hooiveld to put Fulham level.
Asked if he counts that as his first goal for the west Londoners, he smiled and said: "No, I don't. You have to shoot to score and I did, but I think it was an own goal.
"We kept on pressuring them and when Kieran scored I was 100% certain we were going to win this game, but we made stupid fouls to let them into the game."