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 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.
Mourinho handed Brazilian forward Kenedy a first Barclays Premier League start as both managers made three changes and it was Chelsea who had the better of the early chances as Fabregas scooped over from close range before John Terry diverted just wide from a corner.
Nathan Redmond produced the first shot on target when he drew a save from Asmir Begovic with a powerful drive from a narrow angle, while Willian tested John Ruddy at the other end following a threatening run from Eden Hazard.
Redmond was deployed in an unfamiliar central attacking role and was integral to a move that ended with Dieumerci Mbokani lifting a shot over the crossbar from the edge of the box after 20 minutes.
Costa failed to keep his effort down after a slick passing move from the hosts before both teams had penalty appeals turned down as Willian went down under the challenge of Jonny Howson in the box and then brought down Robbie Brady inside his own area moments later.
Chelsea found themselves under pressure in the closing stages of the first half but Ruddy had to be alert to deflect a Costa strike wide following a swift counter-attack.
Kurt Zouma volleyed over from a corner early in the second period before Fabregas forced Ruddy into a diving save with a low drive.
Chelsea's persistence paid off when Fabregas' quickly taken free-kick over the top of the Norwich defence found Costa, who kept his cool to curl a shot around Ruddy.
Zouma could have double the advantage when he headed against the crossbar from a free-kick soon after, while a fierce Brady strike was touched wide by Begovic.
Ruddy kept out Nemanja Matic in the closing stages as Chelsea claimed an important victory to take their Barclays Premier League points tally to 14 for the season, leapfrogging Norwich in the process.
Chelsea manager Jose Mourinho: “Me and the fans didn't deserve to have our heart in our hands in the last four minutes. It is the pressure of the bad results, that is normal, but they coped well. The only thing we didn't do was score more from the chances we had. We should have scored at least two and breathe easier but it is difficult for us to score goals and to keep a clean sheet against a team who plays defensively."
Norwich City manager Alex Neil: "I am disappointed. I thought our players worked really hard, I felt we gave Chelsea very little and we worked really hard. We put some fresh legs on as the game went on, and tried to create problems. Arguably we could have done more but I am happy with my players. We've done it at Liverpool, at Manchester City and here - gone to the so-called bigger clubs and put on a show. It is just disappointing we had that one lapse of concentration."
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