The Belgian gave the hosts the lead after 25 minutes, latching onto Mateo Kovacic's pass to fire low past Alisson.
Hazard ought to have sealed the three points when he was put through by N'Golo Kante in the second-half but the Brazilian keeper kept out his one-on-one effort.
Sturridge made Hazard pay for his wastefulness as the Liverpool striker came off the substitutes' bench to score a brilliant equaliser with one minute of normal time to play.
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Hazard is the Premier League’s top scorer this season with six goals, and the 27-year-old has scored those six goals from just nine shots on target.

Eden Hazard

Image credit: Eurosport

But the Blues' talisman concedes his side missed a golden opportunity to leapfrog their opponents into second place.
"I am a bit disappointed I didn't score in the second half," he said.
"When you are winning 1-0 and concede a goal at the end you can be sad but I think the draw is the fair result.
"I think we can challenge against Liverpool, with [Manchester] City too. These are the best teams in the country. We are not far, we are winning games and confidence is high."
Chelsea manager Maurizio Sarri reiterated his belief that Hazard is capable of scoring 40 goals this season, and it would appear to be getting a response from his star player.

Maurizio Sarri has praised his Chelsea players for their performance on Saturday evening

Image credit: Getty Images

The Italian underlined his satisfaction in his side's performance despite the late setback, believing that the draw was the right result.
He said: "We have started only two months ago I think [we are doing] very well. Maybe they [Liverpool] are one step ahead but I am starting to think we can recover in one season.
"I am disappointed for the moment they scored but I am really happy with the performance. We played against a top level team and we played very well with quality and intensity.
"I have the feeling we are improving match by match. I think we have the quality in the offensive phase. I think the players have been very intelligent to learn so quickly the offensive phase."
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