Pickford also stopped Alvaro Morata after the break as Sarri's men ramped up the pressure on their Merseyside opposition.
Accordingly, the Italian football specialist has broken the record for longest unbeaten run from the beginning of a managerial career in the Premier League, with 12 unbeaten games.
Hazard missed three games with a back injury but played a full 90 minutes for the first time since October 20 during a fractious goalless draw with Everton on Sunday.
"I think we have to sort out the problem of the approach of the match", said Sarri. We were a bit unlucky.
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Chelsea legend Clive Walker wants Maurizio Sarri to use the global break to prepare for the busy fixture schedule coming up, which he knows the Italian is unused to.
But the former Napoli boss does not believe that the two points dropped will prove vital. "When I'm on the ground, I get free kicks and each time it's a chance for Chelsea to score". I know how important it is for Liverpool to win the Premier League.
Everton did well to restrict space for the hosts, with Jorginho particularly ineffective, and Marco Silva revealed that stopping Chelsea as a team, and not just focusing on Hazard, was the crucial factor in claiming a share of the spoils. For example, in the last match in the Europa League [against BATE], we played 81 balls in the opponents' box, but we created only four goal opportunities.
"We came here to stop Chelsea as a team".
The match was often ill-tempered, with Jorginho and Marcos Alonso fortunate not to be sent off for unsafe tackles on Gylfi Sigurdsson and Theo Walcott respectively as well as frequent angry exchanges between the Chelsea and Everton benches.