First time this season as Chelsea to a vital victory
Rudiger and Cesar Azpilicueta were superb in defence.
Luiz blocked as Leroy Sane shaped to shoot after drifting from left to right and Chelsea were struggling to escape their own half.
Hazard after 20 minutes gestured towards the touchline, appearing unhappy at being asked to press the City defence more closely. Sarri voiced his frustrations with Hazard, too.
City had another opportunity. Sterling sauntered by Marcos Alonso down the left and the ball found its way to Sane, whose effort was blocked by Azpilicueta.
Chelsea then struck on the counterattack. Luiz’s long ball found Pedro, whose cross-field ball to Willian did not create the intended shooting chance.
Willian found Hazard, who slid the ball through for the late-arriving Kante to blast in for his second league goal of the season.
Willian had two chances early in the second half. Ederson parried Willian’s free-kick after Fernandinho’s mis-timed challenge on the Chelsea forward.
And Willian shot weakly at the City goalkeeper soon afterwards after Kante’s fine run and pass.
Guardiola reacted, sending on Gabriel Jesus for Sane, and his team pressed Chelsea deep into their own half once more.
Still Chelsea would not relent. A Kyle Walker free-kick was saved by Kepa after Hazard was penalised for handball.
Pedro dragged a shot at Ederson on a rare foray forwards and Chelsea continued to stand firm – before scoring a second.
Substitute Ross Barkley’s shot went wide and a corner was awarded, although replays later suggested there was no deflection and it should have been a goal kick.
Hazard’s inswinging set-piece was met by Luiz, who rose highest to send a looping header into the far corner of the net.
Barkley headed wide for Chelsea and Fernandinho fired off target for City before Jesus was denied by Kepa in stoppage time.