It was an even first period, with good chances from Chris Brunt and Costa coming close to opening the scoring, but to no avail.
The majority of the second half was quiet from either side, and looked to be heading for a draw before Costa opened the scoring in the 76th minute. Cesc Fabregas had one final opportunity but it was the Spaniard's strike that proved to be the difference.
Tit for tat
The Blues went into this clash having been knocked off the top, after Arsenal's 3-1 win over Stoke City and were shaken early by Tony Pullis' side.
Their first opportunity came in the 19th minute, when Brunt decided to let one rip from distance which had Thibaut Courtois scrambling but was a yard wide of the mark.
They had another sniff in the 26th minute when David Luiz had given the ball away to Salomon Rondon. The striker managed to make his way into the area, and flashed the ball across the six-yard box but nobody was there to stick it home.
The home side's best opportunity didn't arrive until the 38th minute, and were whiskers away from drawing first blood. A short corner came to Eden Hazard whose cross looked for Costa, it just missed the strikers head and was also just inches wide of the far corner.
The two sides didn't have the same energy in the second period, with the two sides just having sniffs of an opportunity as the clocked ticked down.
Luiz looked to change the score just after the hour mark, when he fired a free-kick at Ben Foster who tipped it wide but referee Mike Dean wrongly denied Antonio Conte's side a corner.
The Baggies had a second bite of the cherry in the 66th minute, when it looked like their counter-attack had been cleared by Gary Cahill. However the skipper's clearance to Willian was pinched by Matt Phillips, his cross looked to find Rondon but Cahill was there once again to clear.
Willian looked to make up for that incident as he looked to drive at goal, but West Brom showed their defensive solidity as the Brazilian was forced to shank his effort well wide.
A crucial goal
It looked like that the game would be heading for a goalless draw, but a defensive error saw Costa give the home side a crucial lead.
It was an excellent effort from the striker, as his tenacity forced Gareth McAuley into an error on the right wing. He picked the ball up as he stormed his way into the area, and provided a clinical finish as he clattered the effort into far corner giving Foster no chance.
He was it again in the 88th minute as a defensive error from a throw came to the striker. He pulled the ball back to Fabregas as he looked to kill the game, however Craig Dawson's pressured forced the error out for a goal kick.