Striker Costa looked fired up as he celebrated his second goal after ending a six game goal drought in which his form and performances have dipped.
But Conte was full of praise for Costa and his players after their victory put them seven points clear of rivals Tottenham which turns up the pressure on Mauricio Pochettino ahead of their trip to Crystal Palace on Wednesday.
Blues boss Conte said: “I think that we passed a big, big step. A big psychological step after the defeat against Man United. Because, against United, we lost three points and then you had to prepare a semi-final against Tottenham.
“And then another tough game in the league. I think that, in that moment, we have had a really important test, psychological test, and our answer was very good. For this reason I must be pleased. But we must think this race is open.
“We reached the final in the FA Cup, which is a great incentive for us. But the other competition, the league, is totally open.
“But I think we are ready to fight until the end. Then we must be proud for our job, for what we are doing this season. But now it’s normal. It’s normal that we have to try to push until the end to keep this position.”
Conte also praised Costa and said: “I’ve always said I’ve been pleased with his commitment, his work for the team. He always worked for the team. Sometimes he was unlucky in different circumstances, but I was always confident about him.
“He’s very important for us. His confidence is very important for us. He scored two very beautiful goals, good combination and technique between him and his team-mates.”
“It’s important to continue to stay together and to understand this moment, and be ready to push our team in the right way. For this reason, I want to say thanks to our fans. They showed great passion and pushed a lot during this game.”
Chelsea defender Gary Cahill, who put his side back in front in first half injury time, added: “It does feel like a massive step. It’s the first time for a long time we’ve played first [before Tottenham in the league].
“This result leaves us in a good position. We have a big game this weekend [at Everton].”
Saints boss Claude Puel added: “”It was a good game to watch on the TV. It’s a harsh result. It’s important to finish with good performances and stay in the first part of the table.”