Costa netted after seven minutes and Eden Hazard struck a second before half-time the Foxes floundered at the home of their predecessors as champions.
Both goalscorers celebrated by paying tribute to Willian, who is on compassionate leave following the death of his mother.
David Luiz hit the crossbar with a free-kick as Chelsea dominated, with former Foxes midfielder N'Golo Kante excelling, before Victor Moses struck a late third.
Chelsea head coach Antonio Conte laughed off suggestions he was about to be sacked after some bookmakers suspended betting on his departure on Thursday.
Claudio Ranieri knows what it is liked to be fired by Chelsea, so does Jose Mourinho, who brings his Manchester United side to Stamford Bridge on October 23 on Sunday week
That will be another test for Conte's side.
The odds of Ranieri leaving Leicester may be shortening after a start to the season which has seen the champions collect eight points from eight games
Chelsea now have 16.
Chelsea were Leicester's predecessors as champions, under Mourinho, before it all spectacularly unravelled last season into a 10th-placed finish
Twelve months on, Leicester find themselves in a post-title malaise.
The Foxes were robust last season, but have now conceded a league-high five goals from corners this term.
Costa was unmarked at the back post when he met Nemanja Matic's flick of Pedro's corner to score in the seventh minute.
Conte believes hard work and patience are required in a Chelsea revival that will take time.
He wants selfless players in the mould of Kante, who turned down Champions League football with the Foxes to move to Chelsea.
Conte restlessly called upon his team to keep pressing Leicester, with the intelligence, simplicity and athleticism of Kante to the fore.
The back three system Conte has employed worked well with the Blues dominating possession.
And Luiz struck the crossbar with a dipping free-kick after Robert Huth was booked for a heavy challenge from behind on Hazard, who was at his vibrant best.
Chelsea's second was a result of a lucky break, earned through their persistence and pressing of Leicester in the visitors' own half.
Matic fed Pedro, who fell
The ball broke for Hazard, who rounded Kasper Schmeichel to score.
Jamie Vardy touched a Luis Hernandez cross across goal, but still in play, as Leicester tried to respond.
Pedro appealed for a handball in the Leicester box in the opening move of the second half and moments later a bottle was thrown from the away end after the visitors' Jeffrey Schlupp went down looking for a penalty.
Kante almost laid on a third for Chelsea, leading a breakaway and finding Moses with an early pass.
Moses shot first time, but Schmeichel was equal to the effort.
Next Wes Morgan blocked as Kante came close to capping a supreme performance with a goal.
Costa appeared at one moment to call for himself to be substituted and was ignored and Chelsea's third followed.
Moses cut in from the right, received a back-heeled return pass from Nathaniel Chalobah and tucked the ball under Schmeichel.
Source : PA