Costa’s 41st-minute strike was all the Blues had to show for their efforts at the Riverside Stadium with Boro goalkeeper Victor Valdes in particularly stubborn form.
But it was sufficient to secure a sixth successive league win and clean sheet in front of a crowd of 32,704.
Chelsea have not conceded in the competition in 590 minutes and lead Liverpool and Manchester City by a point while Boro, despite their recent heroics away to Arsenal and Manchester City, have won only once in 10 league outings.
Antonio Conte’s men were laboured in the early stages as the home side, far from trying to soak up pressure, decided attack was the best form of defence.
Having already stretched the Blues’ rearguard on two occasions, they threatened when full-back Fabio played Gaston Ramirez in down the left and although Marten de Roon’s initial shot was blocked by Nemanja Matic, the ball dropped kindly to Alvaro Negredo, who fired it across goal.
Chelsea gradually worked their way into the game with wingers Victor Moses and Marcos Alonso seeing plenty of the ball, but with Eden Hazard well marshalled and Costa struggling to find space, they were not causing genuine problems.
But as the pressure mounted, Boro finally wilted and, but for a fingertip save by Valdes from Pedro, the visitors would have been in front with 28 minutes gone after Hazard had lifted a superb ball over the top into Moses’s run.
Antonio Barragan and Fabio combined to block Pedro’s effort nine minutes later with Valdes repelling Alonso’s follow-up, and Moses saw another effort deflected wide by Fabio, but it was from the resulting corner that Chelsea took the lead. Alonso’s header cannoned off the unwitting Ramirez’s back and fell nicely for Costa to stab home from close range.
Negredo forced goalkeeper Thibaut Courtois into a vital save after Traore had found him with 12 minutes remaining, and Boro finished empty-handed.
Middlesbrough: Valdes, Barragan, Chambers, Gibson, Da Silva (Downing 71), Traore, de Roon, Clayton (Fischer 73), Forshaw (Leadbitter 89), Ramirez, Negredo. Unused substitutes: Bernardo, Rhodes, Guzan, Nsue.
Chelsea: Courtois, Azpilicueta, Luiz, Cahill, Moses (Ivanovic 90), Kante, Matic, Alonso, Pedro (Chalobah 80), Costa, Hazard (Oscar 90). Unused substitutes: Begovic, Fabregas, Batshuayi, Terry.
Referee: Jon Moss (W Yorkshire).
Man of the match: Victor Valdes.