The Spain international turned home his 10th top-flight goal of the season from a corner late in the first half to secure a sixth win and clean sheet in a row for the division's form side.
Chelsea scored 16 goals in their previous five league games but were kept at arm's length early on by Aitor Karanka's disciplined side, with Victor Valdes' save from Pedro the only opportunity of note.
Boro have held Arsenal and Manchester City to draws this season, but just as belief in their chances of another upset were growing, Costa took full advantage of some poor defending to open the scoring on 41 minutes.
The home side put up a spirited fight in the second half but struggled to carve out clear-cut opportunities, and Chelsea themselves wasted two excellent chances through Pedro and Victor Moses.
Alvaro Negredo was denied by Thibaut Courtois in the closing stages but Chelsea nonetheless cruised through relatively untroubled, as they moved a point clear at the top ahead of City and Liverpool, while Middlesbrough stay 15th with 11 points.
Eden Hazard was well marshalled in the early stages, but one superb chip allowed Moses to cut the ball back for Pedro 12 yards from goal, only for Valdes to tip his former Barcelona team-mate's shot over the crossbar in brilliant fashion.
Adama Traore was Middlesbrough's biggest threat throughout and one positive run at the visiting back three allowed Antonio Barragan to cross for Negredo, who headed just wide.
Boro had frustrated Chelsea in open play, but they were undone by a mistake from a set-piece just four minutes from the break. Hazard's corner deflected up into the air off Calum Chambers and Costa reacted quickest to poke the ball beyond Valdes.
Chelsea almost had a second within the first minute of the second half, Valdes making a good low stop to deny Marcos Alonso before Chambers prevented Pedro from turning in the rebound.
The hosts responded with their best spell of the match, with Gaston Ramirez blasting a half-chance over from 20 yards, and they were dealt a slice of luck when Pedro's half-volley from Costa's knock-down crashed against the crossbar and bounced out.
Moses skied another good chance on the counter-attack after a clever Costa throughball, but Chelsea rarely looked likely to be made to pay for their second-half profligacy, with Negredo's half-volley beaten away by Courtois in Boro's best opening.
The win keeps Chelsea's momentum going ahead of a potentially crucial period, with a home clash against Tottenham next week followed by a trip to Pep Guardiola's City.
KEY OPTA STATS:
- Diego Costa became the first player to reach double figures for Premier League goals this season (10 goals).
- Indeed, Costa has scored in six of his last seven Premier League away games, netting seven goals in total in that run.
- Only Swansea (2) have won fewer points at home in the Premier League than Middlesbrough this season (4), while no side has scored fewer top-flight goals at home than the Boro (5, level with Hull and Swansea).
- Adama Traore completed 12 dribbles in this match, more than any other player in a single Premier League game this season.
- Meanwhile, Eden Hazard created six chances for his teammates, his highest in a Premier League game since May 2015 (8 vs Sunderland).
- Middlesbrough have now lost their last six Premier League games against Chelsea, failing to score in each match.