And although they do not appear as strong as Liverpool and Manchester City, Chelsea will always have hope when Hazard is on the pitch.
"I know when I play good, the team play good, so I need to be ready for everything," said Hazard. "The season is long but we will fight to the end and we will keep improving."
Hazard's goal came with five minutes remaining after he played a one-two with Marcos Alonso and sent his shot under the body of former Chelsea goalkeeper Asmir Begovic.
The Blues were already ahead at that point after Sarri's decision to send on Olivier Giroud and Pedro paid off, as the pair combined for the Spaniard to net his third goal of the campaign.
Despite admitting that Chelsea's start under him is going better than expected, Sarri is refusing to take any notice of history.
"At the moment, I'm very happy with my players," said Sarri. "They improved very well during the week and in matches. I'm happy to have 12 points from four matches but I always think we can do more. We can improve."
Asked about Chelsea's record of winning the title when they start with four victories, Sarri added: "I think that now for us, it's better to think about the next match. The gap at the beginning was 30 points. I think it's very difficult to recover this gap in only one season. I think only about the next match. Not the end of the season."
Alonso's contribution was almost as important as Hazard's and Sarri had high praise for the left back who has scored, made two assists and won two penalties for Chelsea this season.
"Alonso at left back is maybe the best in Europe," said Sarri. "He's doing very well in the offensive phase but he can improve in the defensive phase. If he improves that, he could be the best left back in the world."
Eddie Howe's team suffered a first defeat but the Bournemouth manager was heartened by the fact the Cherries should have capitalised on poor defending, with Callum Wilson and Nathan Ake missing from close range, and also had a strong penalty call turned down.
"I thought we did well for long periods," said Howe. "Chelsea are a different team this year, a totally different way of playing. But we nullified their strengths and had the best two chances, chances we didn't take, and we paid the penalty with a cruel deflection off Steve Cook."