Costa's 12th goal of the season sees the Blues maintain their three point advantage over Arsenal -- who briefly went top on goal difference.
Costa's 12th goal of the season sees the Blues maintain their three point advantage over Arsenal -- who briefly went top on goal difference after beating Stoke 3-1 on Saturday -- whilst the Gunners' north London rivals Spurs lost ground after going down to a 1-0 defeat at Manchester United also on Sunday.
Spurs remain in fifth on 27 points, 10 adrift of Chelsea, whilst United -- for whom Henrikh Mkhitarayn opened his account for them in the Premier League before going off on a stretcher in the second-half with what appeared a left ankle injury -- are 13 points shy of the leaders but only six off the Champions League spots.
West Brom made it difficult for Chelsea at Stamford Bridge until Costa seized on the ball 14 minutes from tiem brushed off the attention of Gareth McAuley and rifled the ball past Ben Foster into the top far corner.
Chelsea manager Antonio Conte celebrated the victory in his trademark emotional manner and afterwards feted his players for their winning nine in a row -- scoring 23 and conceding just two goals to boot -- in a 'very tough league'.
The Italian -- who made his name as a coach at Juventus for whom he also played with distinction -- saluted their character as well in remaining focused and patient.
"I'm very happy. We faced a tough, physical team. They're difficult to break down and we showed great patience to find the right solution," said the 47-year-old.
"It wasn't easy but my players showed me great character and motivation to win.
"We are working very well. When you don't find the goal, it becomes more difficult but we were very good to find in the right moment to do it."
The United and Spurs match was of a far better quality with the hosts going close through Paul Pogba, who hit the post and also forced Spurs' French goalkeeper Hugo Lloris into a superb save in the second-half.
The visitors could have gone home with a share of the points but for a woeful mis-directed header from unmarked Kenyan midfield enforcer Victor Wanyama.
In a match involving two struggling teams Southamtpon ended a bad week with a much-needed win, beating Middlesbrough 1-0 -- record signing Sofiane Boufal securing the three points with a delightful effort from the edge of the area.
Victory -- only Southampton's second in seven matches -- moves Saints eight points clear of the relegation places whilst 'Boro remain mired in the battle for survival, just three points better off than third from bottom West Ham.