Costa ended a six-match barren spell, prompting chants of "we are staying up" and "Jose Mourinho", as the Barclays Premier League champions secured a fourth win in their 13th Barclays Premier League game of the season to edge away from the foot of the table.
Three years to the day since Rafael Benitez was installed as interim successor to Champions League-winning boss Roberto di Matteo, Mourinho was desperate to end a run of three straight league losses in front of the watching Roman Abramovich.
Mourinho, banned from the stadium for the defeat at Stoke, was the subject of the first managerial vote of confidence in Abramovich's 12-year ownership on October 5.
The caveat of the board's support was that results had to improve and Chelsea have now won three times in eight games since
Here, Costa converted Cesc Fabregas' quick free-kick for just his fourth goal of the season and first since last month's defeat of Aston Villa.
Costa had earlier been conspicuous by his absence when Eden Hazard - some of his effervescence restored - twice crossed well, while he wasted two other chances, reminiscent of Fernando Torres.
He blasted over Pedro's cutback and then was denied by John Ruddy two minutes before the break.
Kurt Zouma also hit the bar after Chelsea had gone 1-0 up.
Norwich had opportunities, with John Terry blocking a Sebastien Bassong effort and Asmir Begovic saving well from Robbie Brady.
And, as with many Chelsea matches this season, there were penalty controversies.
Costa and Bassong grappled at one point and Willian went down under a Jonny Howson challenge before escaping a foul for bundling over Brady at the other end.
Mourinho selected 19-year-old Brazilian teenager Kenedy at left-back and restored Fabregas to his starting line-up.
The Blues' attacking build-up play was improved, but they were defensively suspect and Nathan Redmond was denied by Begovic after Terry's poor headed clearance.
Fabregas had fired a Hazard cutback over by then and Willian was denied by Ruddy before Costa's first gilt-edged chance.
Referee Craig Pawson rejected three penalty appeals - two from Chelsea, one from Norwich - in the first half, with replays justifying the decisions.
The match could have swung either way in the space of 60 seconds prior to half-time.
Terry blocked a Bassong shot as Chelsea scrambled clear a corner, with Kenedy crossing and Costa's effort deflecting wide off Ruddy's right leg on the counter.
Hazard was growing in confidence as Chelsea pressed in the second half, but few of his team-mates were on his wavelength and Costa appeared on a different frequency altogether.
Another Hazard cross evaded everyone, with Costa absent.
He was present when it mattered, though, bringing down Fabregas' free-kick before cutting inside Ryan Bennett and shooting inside the far post.
Zouma directed a Willian free-kick on to the bar soon afterwards and Nemanja Matic waltzed through only to be denied by Ruddy.
Norwich had chances, too, with Brady's shot saved by Begovic, but Chelsea held on to ease the pressure on Mourinho, who punched the air in relief when he heard the final whistle.
Source : PA