كتب : وكالات
As hosts City took a first-half lead through Stevan Jovetic, Chelsea boss Jose Mourinho sent in Salah for Samuel Eto'o at half time.
Even though the Egypt forward looked lively for long spells, France midfielder Samir Nasri settled it with the second goal.
Chelsea have now been knocked out of both cup competitions in England but remain on course to win the league and Champions League double.