The Blues presented Bournemouth in their fourth Premier League-game of the season.
After the Cherries had made a resolute defense, Chelsea finally managed to break them when Pedro scored with another 20 minutes to go.
Hazard provided five minutes before time for a result with a fine finish to make it 2-0.
The Belgian now scored twice and assisted twice in two starts in the season this season.
And Sarri never doubted who his main player was during his press conference after the game.
"I do not know if we can win anything this season, but Eden Hazard is very important to us," Sarri said.
"In the first half it was very difficult to play – we had to move the ball at a higher speed – but it was not easy.
"We needed more exercise without the ball, but I think we did well.
"I am really happy with my players at the moment, I do not want to think about the last four matches – it's better to think about the next one."
Hazard has been constantly associated with a relocation from Stamford Bridge this summer after heavy interest from Real Madrid.
The equipment from West London, however, resisted the European champions after he announced that the 27-year-old was not for sale.
And the former forward row from Lille has encouraged Chelsea fans to stick after he has revealed that he is happy with the new boss.
He said: "I am confident, I enjoy my football, we have a new manager and many good players.
"I know when I play good the team is playing well, so I have to be prepared for everything.
"The season is long, but we will fight to the end, and we will continue to improve."