Fonte, who was named in Opta's Premier League Dream Team earlier this week, told BBC Sport: "We started the game very well.
"The first 20 minutes was very strong - we controlled the game. But after that we went to sleep a little bit and they scored.
"But we managed to stay in the game, had a good team talk at half time, and in the second half we came back and kept going, kept pressing. We dominated the second half and deserved the equaliser."
Southampton's point saw them climb up to fourth in the Premier League table with 15 points.
The Saints currently boast the division's best defensive record, having kept five clean sheets and conceded just three goals in eight games.