Chelsea moved to within five points of fifth-placed Manchester United with a 2-1 win over Norwich.
The Canaries were undone by their own defensive failings – conceding after 39 seconds to Kenedy’s fastest goal of the Barclays Premier League season so far – and misfortune, as the victims of an incorrect decision by the match officials.
Costa’s strike before half-time should have been ruled out for offside, but assistant referee Michael McDonough failed to flag and the goal stood, despite protests as Norwich lost for the seventh time in eight Premier League games.
The Canaries ensured Chelsea never truly had a grip on proceedings. Cameron Jerome hit the bar early in the second half and Nathan Redmond pulled a goal back 22 minutes from time.
But Alex Neil’s men dropped into the relegation zone as Chelsea climbed to eighth with a third straight league win.
Guus Hiddink remains unbeaten in domestic competition as Chelsea interim boss and Costa has now scored 10 goals in 14 games under the Dutchman.
Chelsea made four changes following the win at Southampton and Norwich made one of the incomers particularly welcome in the opening minute.
Eden Hazard ran at Norwich and switched ball left for Kenedy, who cut in and was allowed to make space to shoot. After Ivo Pinto, Alex Tettey and Ryan Bennett failed to shut him down, the Brazilian fired a low effort into the bottom corner across goal.
Chelsea threatened to score another, with Hazard revelling in the open start. The Belgium playmaker pulled the ball back for Bertrand Traore, on his first start, but it was just behind the forward.
The game continued to be open and Jerome headed the ball on to the roof of the net from Wes Hoolahan’s cross.
Gary O’Neil came on for Tettey, who hobbled off after an ineffective 36 minutes, and provided more stability, but still Norwich were vulnerable.
Hazard tried to fashion a second for Traore, but the 20-year-old, who had scored on his last three substitute appearances, took a heavy touch and John Ruddy picked it up.
Norwich supporters were incensed when Thibaut Courtois picked up the ball soon afterwards, from Gary Cahill’s pass-back. What the home faithful did not see, but referee Lee Mason did, was that the ball deflected off Jerome.
The Canaries had a more legitimate reason to feel aggrieved when Cesc Fabregas took a quick throw to Traore and he found Costa, who peeled round to collect the ball before clipping it over Ruddy. Replays showed Costa was clearly offside.
Jerome spurned a fine opportunity to pull a goal back when Hoolahan crossed and Russell Martin headed the ball back in front of goal. Jerome was unmarked when, with the ball at a difficult height, he swivelled and hit it left-footed on to the bar.
Hiddink sent on John Obi Mikel for Oscar and Willian for Traore.
But the control he sought was lacking and it was soon 2-1 as Redmond ran in behind Kenedy and latched on to a Hoolahan pass before rifling past Courtois.
Kenedy was immediately withdrawn by Hiddink, with Baba Rahman back into the fray following his half-time substitution at Southampton.
Redmond twice went close and Costa slid the ball wide at the other end as Chelsea held on.