Hazard told Sky Sports before the game that it was a chance for Chelsea to strike a mental blow against their title rivals Tottenham, to follow the FA Cup semi-final win by extending their advantage at the top of the Premier League to seven points and "create a bit of trouble in their minds".
On Wednesday, Tottenham travel to face a Crystal Palace side whose win at Stamford Bridge earlier this month did much to reopen a title race that Conte will hope results such as this wrap up swiftly.
Conte, whose side had been beaten twice in their past four league games, said: "We passed a big step - a big psychological step - after the defeat against Manchester United".
"Our answer was very good".
"I must be pleased because I have a squad of great men and great players. But we fight until the end".
While the front two looked sharp, the rest of the Chelsea team were struggling to assert their control over Southampton and the Saints deservedly equalised midway through the first half to make it 11 matches without a Premier League clean sheet for the hosts - their worst run in more than 20 years.
Conte reckons players can only be great by playing as part of a team.
"This season it's very important to understand that it's not always about who spends more money who wins".
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"For the forward it's very important to score. because the goal is your life", Conte said.
Ryan Bertrand pulled a goal back for the visitors in second-half injury time, heading home nicely past Thibaut Courtois for Southampton's second on the evening, but the game was in hand for Chelsea by that point.
Southampton playmaker Dusan Tadic forced Courtois to save sharply before the hour and Conte, typically buoyant when Costa netted, was kicking at his technical area when Yoshida and Gabbiadini again went close from a poorly defended corner. He recovers the ball so well and is a technical player.
Terry, who has won 14 major trophies at Chelsea, could finish the season lifting more silverware and received acclaim from the home faithful.
Conte said: "He knows my admiration for him". To become a great or fantastic season there is only one way: to win [the title].
Kyle Walker, Danny Rose, Toby Alderweireld and Eric Dier are all reported targets, while former Barcelona midfielder Xavi has suggested Dele Alli is the type of player his old manager Pep Guardiola would like to sign for Manchester City.
And Hazard nearly made it two on the counter-attack in the 17th minute from another Costa pass, but his first-time shot sailed well over the crossbar from just outside the penalty area.
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