But fellow England worldwide Barkley marked his return to the national team with his first goal for Chelsea - nudging in Oliver Giroud's inventive cross with his shin.
Hazard played in substitute Alvaro Morata to seal the win in stoppage time.
It was indeed the Belgian who played the final killer pass to Morata, who finished with aplomb for his second league goal of the season.
"Maybe I need to think about that, some people only think about goals".
He told Sky Sports: "We played a very good match".
"I think it's down to us to work hard and keep improving as much as we can in training - even though we're not playing - and to try and stay sharp because you might get an opportunity".
But despite Sarri's caution, England global Barkley is confident Chelsea can rival Manchester City and Liverpool this term.
In the second half, Ross Barkley rose to the occasion and made it 2-0 for his side after receiving a lay off from Olivier Giroud. If this is to be a three-way title tussle - it is increasingly looking that way - then the Blues could have asked for little more than an away win while both City and Liverpool went on to drop points.
"I want to keep doing it and keep everyone happy".
They will look to carry their fine form on following the domestic pause when they take on under pressure Jose Mourinho's Manchester United.