The promoted side were a handful for Chelsea and caused them problems in the first half with Riyad Mahrez forcing Thibaut Courtois into a good stop.
Chelsea took over after the half-time break, and Eden Hazard followed up Costa’s opener with a deflected effort.
The outcome could have been different had David Nugent not wasted a fantastic one-on-one opportunity in the second half, but his shot was well saved by Courtois.
After a lacklustre first half, Chelsea sharpened up in the second and the Leicester City goal came under a lot of pressure as Oscar hit the base of the post and Branislav Ivanovic forcing a fine stop from Kasper Schmeichel.
Jose Mourinho’s men eventually over-powered the promoted side as Costa provided the finish to Ivanovic’s good work and Eden Hazard secured the win.