The Spaniard broke the deadlock shortly after the restart, before Eden Hazard made sure of the points in the closing stages.
Here, Sports Mole looks back over the contest to determine if the result was a fair one.
On target: 10
On target: 4
Was the result fair?
The Premier League is a ruthless and at times cruel division and Leicester found that out today. They arguably created just as good a chances as Chelsea did, but the hosts converted theirs, while Leicester fell short in this area. They will feel a little hard done by, but Chelsea were deserving winners.
Jose Mourinho is bound to have been disappointed with what he saw during the first half because his side were pedestrian in possession. That changed after the restart. There was much more purpose about their play and the passes started to stick. The goals also arrived, but he will be slightly concerned that defensively at times Thibaut Courtois had to be relied upon when Leicester were presented with clear-cut chances that they probably should have scored.
Leicester will not be judged on this result, but Nigel Pearson will be able to take some positives despite the defeat. His side were well organised throughout, but perhaps even more pleasing for the Leicester boss was the fact that his players showed little fear. They attacked Chelsea and created some really good openings. If they play like this against some of the lesser sides in the Premier League, they should pick up some victories soon.
Sports Mole's man of the match
Thibaut Courtois: It's not often that a goalkeeper will get the award when his side has won 2-0, but the Belgian made two crucial saves just before Costa went on to score the game's opening goal. Aside from that, he relieved pressure on his defence time and again by coming out to claim crosses that were hoisted into his area.
When you go away to teams like Chelsea, you cannot afford to pass up goalscoring opportunities. Moments after he had spurned a good chance, David Nugent fluffed a golden one. He was put through clear on goal, but with just Courtois to beat, the Leicester striker appeared to hesitate and when he did eventually shoot, it was weak and ended up being an easy save for the Chelsea goalkeeper. To make matters worse, just a couple of minutes later, Costa broke the deadlock up the other end.
There was not too much for Lee Mason to do in all honesty, but he allowed the game to flow throughout and was sensible with his decision making. Chelsea did have a couple of penalty shouts, but the man with the whistle was right to wave away the appeals.
Chelsea: Next Saturday, Mourinho's men will travel to Merseyside for a clash with Everton at Goodison Park.
Leicester: Meanwhile, Leicester will return to the home comforts of the King Power Stadium a day later, with Arsenal the visitors in the Premier League.