Hazard has been Chelsea's standout player this season by some distance, contributing 10 goals and nine assists in 20 Premier League games.
But he was helpless to improve Chelsea's fortunes at Stamford Bridge on Wednesday, as a limp collective performance saw them drop points for a second successive home game.
Hazard was lively, though no one else in a Chelsea shirt appeared to be playing on the same wavelength.
Kante acknowledged Hazard's performance, but says the rest of the team has to do more to contribute.
"We all know Eden is a fantastic player, he's helped us a lot during this season and the seasons before," Kante told Chelsea's official website.
"He can make the difference at any time, but we need to count on everyone. We created chances that we missed. It's a difficult moment but we need to bounce back."
Wednesday's draw saw the Blues have an Alvaro Morata goal ruled out due to offside, while Southampton goalkeeper Angus Gunn also denied them multiple times.
Kante attributed the result to a bit of poor fortune and hopes the club can improve in their efforts over their upcomings cup matches in the next week.
"We were a little bit unlucky because that was the kind of chance from which we can score," Kante said.
"Maybe we need to find the right timing, but the game was a draw and now we need to bounce back in our next match, in the FA Cup, and in the semi-final of the Carabao Cup. We will see what we can do this week."
The draw also saw another injury setback for the Blues as Willian was forced off in the first half, replaced by Ruben Loftus-Cheek.
He joins an injury list that includes Pedro, Olivier Giroud and Callum Hudson-Odoi, during a part of the season where the games come fast and furious.
Chelsea face Nottingham Forest on Saturday in the FA Cup, have the first leg of their semi-final Carabao Cup tie against Tottenham next Tuesday, followed by their next Premier League contest that weekend.
However, despite those loses, Kante says they cannot be used as excuses with improvement over those fixtures necessary.
Kante added: "It’s difficult because at this time of the season we’ve lost a lot of players, but we still have a good group and I think we need to do better in the next few games."