It looked like it was going to be a dull first period, with half-chances from Pedro, Marco Alonso and Alvaro Negredo. However a volley from Costa in the 41st minute proved to be the difference.
Decent chances followed from both sides in the second half, with Alonso, Gaston Ramirez, Pedro and Adama Traore having their opportunities, but Antonio Conte's side held on for their sixth consecutive league victory.
Off to a very slow start
The opportunity was there for the Blues to go top when they arrived on Teesside, with Liverpool and Arsenal both drawing the day before. However, you wouldn't have thought that as both sides started very slowly at the Riverside Stadium.
Victor Moses and Alonso had half decent opportunities, but the first test of a goalkeeper didn't come until the 28th minute. The ball was chipped over to Moses on the right flank, and the Nigerian did well to pull it back to Pedro. The Spaniard's first-time effort looked difficult to keep out, but Victor Valdes did well to tip the effort over.
Boro had an opportunity of their own in he 36th minute, Traore had impressed over the past few weeks and was it again as he burst past N'Golo Kante and played the ball in. The cross was cleared as far as Negredo, who was seemingly distracted by the jump of Ramirez, as the striker put his header wide of the mark.
Costa puts visitors ahead before the break
It looked like that both sides would be heading into the break goalless, but seemingly out of nowhere the Blues were ahead. It was poor on the home side, a seemingly harmless corner to the near post was deflected into the air, and Costa was the only one to react as he got between three defenders to volley home.
Both sides flying out of the traps
It was a lot better from both sides as the second period got underway, and both had opportunities to add to the scoreline. The first came just a minute after the restart, Moses passed inside to Eden Hazard who played it off to Alonso, but his low effort was well saved by Valdes.
It was then the turn of Boro, Negredo gave it quickly to Ramirez on the edge of the area, but his effort just skimmed over Thibaut Courtois.
Game burns out late on
Chelsea came closest to their second goal in the 63rd minute, the ball was clipped up to Costa who headed it down into the path of Pedro, but his first-time strike just crashes off the underside of the bar.
The final opportunity of the contest came in the 78th minute, and it fell to the home side as they looked to get themselves back into the contest late on. Traore was put through and played it off to Negredo, the Spaniard had a great first touch before getting his shot away but the Courtois produced a smart save.