City had scored multiple goals in each of their previous 13 league wins at the Etihad Stadium but only had one against Chelsea, with Silva netting his seventh of the season in all competitions in the 46th minute.
The midfielder capitalised on a long ball fed through the defence by David Silva and cut in front of Marcos Alonso in the box to beat Thibaut Courtois.
Chelsea did not attempt a shot in the first half, with Victor Moses' misguided strike in the 54th minute their first try, and their only late challenge came from a pair of corners in the 80th minute.
N'Golo Kante was ill and did not make the substitutes' bench for Chelsea, who have now lost four consecutive away games in all competitions -- their longest such dip since 2002-03.
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