The Spain international has struggled for form this season but his second-half effort gave the Blues all three points against the Canaries
Diego Costa signalled the end of a three-game losing streak for Chelsea in the Premier League as the champions picked up a 1-0 victory over Norwich City on Saturday.
Since beating Aston Villa 2-0 on October 17, Chelsea had been defeated by West Ham, Liverpool and Stoke City to leave them 16th in the top flight.
However, Costa halted his own barren run of 603 minutes without a goal in all competitions to lift Chelsea spirits at Stamford Bridge.
Both teams had chances to go in front during an open first half, with each having an appeal for a penalty turned down by referee Craig Pawson.
It looked destined to be another frustrating day at the office for Chelsea until Costa capitalised on some quick thinking from Cesc Fabregas to send his side in front after 64 minutes.
The result sees Chelsea climb above Norwich into 15th and manager Jose Mourinho will hope it marks the beginning of an upturn in form.