Defeat for Mark Hughes" side means they remain in 16th, and their run without a win against the Premier League's "big six' is extended to 26 games.
"I want what's good for me, but I want what's good for the club because the club has given me everything", he was quoted as saying in several national newspapers.
"I've dreamt about this club. So we try to move the ball, then when my team-mates see me free, they try to find good passes and that's what we are doing at the moment".
"I don't have time, but we will speak about my future soon, that I believe".
He added: "I am really very happy for our fans because our support today was excellent".
"I don't want to say that I'll sign a new contract and then not sign it".
"I don't know if I've played better [than I am at the minute], I'm just playing good football because we [Chelsea] are playing good football".
Yet while this initially looked as though it would be a stroll for the visitors, Southampton could easily have made it a more uncomfortable afternoon had they made the most of a number of good chances.
The Belgian marvel has taken his game to another level under the guidance of Maurizio Sarri this season, playing with less restriction and devastating precision as he sits atop the goalscoring chart - with seven in the league.
Former Chelsea midfielder Michael Essien hopes Eden Hazard stays at the club and continues his impressive form. The Belgian now leads the way in the Premier League top goalscorers table with six, and Southampton will have find a way to nullify him if they're to come away with anything.
Sarri's side are yet to be beaten in the Premier League this campaign and won their sixth Premier League game this season at St Mary's Stadium over the weekend. I said to him from the first day that he has great quality - physical and technical quality. "I am happy and I don't need anything", he said. He had some bad moments past year with injury, but I think he can reach a fantastic level because he is a fantastic player.
"I had a lot of injuries last season but hat's behind me now and I'm looking forward to the rest of the season".