The 29-year-old scored an 88th-minute equaliser to earn the Saints a point as he cancelled out Emanuele Giaccherini's strike for the Black Cats.
Fonte told the club's official website: "It was a great moment – I'm happy for myself, the fans and the team, 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."
Southampton have taken four points from their opening two Premier League games.