كتب : Sherif Tarek
The Ghanaian international notched the sole goal on 18 minutes after diverting Frank Lampard's free-kick into the net.
Upon claiming all three points, Chelsea advanced to second place with 61 points, trailing leaders Manchester United by four.
Tenth-placed City remain on 35 points, just six above the relegation zone.
The Blues haven't lost since Guus Hiddink succeeded Luiz Felipe Scolari as head coach at Stamford Bridge about a month ago.
Watch Essien's goal