The defender headed in an 88th-minute equaliser after Emanuele Giaccherini had given the visitors an early lead at St Mary's.
The result leaves Southampton fourth in the Barclays Premier League standings with a win and a draw from their opening two fixtures.
But, despite being happy with his leveller, Fonte was left to regret what he thought was a missed opportunity in front of a boisterous home crowd.
"It was a great moment – I'm happy for myself, the fans and the team," he told Saints Player
of his goal, "but we're just disappointed not to get three points at home.
"When you go a goal down after two minutes, you have to run after the result. It was a very hard game and we weren't at our best – especially in the first half – but I believe we deserved more than just a draw.
"We had the better chances and we created the most opportunities, and we had a a couple of penalty shouts," he continued.
"There was one on Adam and one on me in the first half which I think were blatant. I'll have to look at the video to confirm it, but I'm almost 100% sure.
"We had our fans behind us, and we felt we deserved a bit more. Six points [from six] would be perfect."
Saints had started the game slowly, and were duly punished on three minutes when Giaccherini headed Sebastian Larsson's corner beyond Artur Boruc.
And Fonte says he and his teammates will take lessons from that goal, having been given a few stern words from manager Mauricio Pochettino at the break.
"We're disappointed [with the goal]," he admitted. "We watched it at half-time and he was just standing in the middle of the box, and no-one picked him up.
"The manager gave us a great speech at half time. He told us how it is – what we had to do and what we were doing wrong. We knew we had to change something because the tempo was too slow in the first half.
"It's something we need to learn from because it can't happen again."Mix