Costa, signed in the summer from Atletico Madrid, broke the deadlock in the 62nd minute following a scoreless first half that featured few chances for either side. Eden Hazard doubled Chelsea's lead in the 77th minute, before Costa made way for Didier Drogba to sustained applause from the home supporters.
The result puts Chelsea atop the early table with six points from their first two matches with five goals scored and just one conceded. But although they were made to work by a spirited Leicester City, the Blues will be pleased with their early results, especially with Costa scoring in both matches.
Up next for Jose Mourinho's side is a trip to Everton next Saturday in a match that could serve as an early barometer of Chelsea's credentials. The Toffees finished fifth last season and will be hoping to make another assault on the top four this term.
Leicester, meanwhile, have collected only one point from their first two matches back in the Premier League after securing promotion from the championship. But Nigel Pearson's team can take some positives from the match after playing Chelsea evenly for more than an hour and creating numerous chances of their own.
The Foxes' tricky start to the season continues next Sunday with a match at home to title hopefuls Arsenal.
Facing a promoted side for the second time in as many matches, Chelsea again started slowly. But while last week's opener at Burnley saw the Blues go behind by a goal in the first half, this time neither side was able to break the deadlock before half-time.
Chelsea had the first good chance in the ninth minute, Cesc Fabregas slotting a slide-rule pass to Costa's feet along the edge of the box. Costa allowed Andre Schurrle to run on and take possession, and the German sent his shot off target.
The visitors threatened as well, often through the dangerous Riyad Mahrez, who turned Cesar Azpilicueta in the 32nd minute before seeing his low shot saved. Chelsea hit back four minutes later through Costa, whose shot was deflected off target after fine work on the right by Hazard.
But if the first half was light on chances, the beginning of the second period more than made up for it.
Oscar sent the first warning for Chelsea, curling an excellent effort off the base of the far upright in the 48th minute. Moments later, Branislav Ivanovic forced Kasper Schmeichel into a sprawling save with a well-placed header.
At the other end, Thibaut Courtois saved with his legs to keep out David Nugent's low drive in the 52nd minute. The Chelsea goalkeeper made another fine stop three minutes later, keeping out Nugent again after Leicester countered brilliantly.
Fabregas then came close in the 57th minute, turning his marker deftly before trying to chip Schmeichel, who anticipated the shot and saved easily. But by then a goal felt inevitable, and within five minutes, the opener finally arrived.
Oscar threaded a pass through to Ivanovic, who cut inside on the right and crossed for Costa. The latter controlled with his chest and finished under the onrushing Schmeichel to give Chelsea a 1-0 lead.
With his side finally ahead, Mourinho made a move to consolidate the lead by introducing Ramires in place of Schurrle. Another half-chance followed in the 69th minute, Hazard firing straight at Schmeichel from the edge of the box.
Hazard only had to wait eight more minutes for his goal. In the 77th minute, the Belgium international picked up possession on the left wing, cut inside and beat a defender before finishing with a deflected shot to make it 2-0.
The points secured, Mourinho introduced Drogba for the final 10 minutes. The Ivorian enjoyed a brilliant first spell with Chelsea, and his return to Stamford Bridge could signal another period of success for the Blues.
But with Costa serving as the focus of the attack, the results thus far have been exactly what Mourinho and Chelsea wanted.
|Chelsea Player Ratings|
|Ramires (for Schurrle, 63')||N/A||6.7|
|Willian (for Oscar, 69')||N/A||7.3|
|Didier Drogba (for Costa, 79')||N/A||6.0|
|Ratings via WhoScored.com|
|Leicester City Player Ratings|
|Ritchie de Laet||6.8||6.6|
|Marc Albrighton (for Mahrez, 67')||N/A||6.2|
|Gary Taylor-Fletcher (for Hammond, 72')||N/A||5.9|
|Chris Wood (for Ulloa, 83')||N/A||6.3|
|Ratings via WhoScored.com|
Chelsea manager Jose Mourinho (via PremierLeague.com):
We didn't play good in the first half. Leicester came with great defensive intensity, trying to scare us in counter-attack and we were a bit slow—slow in moving the ball and slow thinking and we needed to change and we did that.
The second half was compete control. I don't forget they had a good chance with the result 0-0 but we had so many chances, so many opportunities to score obviously we deserved the victory.
Leicester City manager Nigel Pearson (via PremierLeague.com):
We're disappointed. Key moments didn't quite go our way today. I thought we played exceptionally well in the first half but it's about taking your chances when you get them.
There are more positives today than there are negatives. We're not talking about us coming to places like this and being happy to lose the game, but I think the players showed what they're capable of. What we need to do is sustain it for the entirety of the game.