The 25-year-old Spain striker, who cost the Blues £32million when they signed him from Atletico Madrid earlier this summer, grabbed the opening goal of the game from six yards out in the 63rd minute on Saturday after good work down the right by Oscar and Branislav Ivanovic.
That took Costa's goal tally in a Chelsea shirt to six, having netted against Olimpija Ljubljana, Fenerbahce and Real Sociedad (two) in pre-season and in the 3-1 win at Burnley last Monday night.
Eden Hazard doubled the home side's advantage on 77 minutes when he cut into the box off the left wing after a Cesc Fabregas pass and fired in a shot that was deflected past Kasper Schmeichel by Wes Morgan's heel.
Leicester, back in the top flight for the first time since 2003/2004, had proved hard to break down and could have taken the lead when David Nugent was put clean through, but Belgium goalkeeper Thibaut Courtois - again picked ahead of Petr Cech - made a smart save.
Both sides were positive during the opening stages, and Leicester, playing in red and having made a couple of changes from their opening-day 2-2 draw with Everton, were not looking in awe of a side expected to contest the Premier League title once again.
A deflected shot from Oscar looped up over Schmeichel and onto the roof of the net before World Cup winner Andre Schurrle drove his 20-yard effort high into the Shed End.
Leicester, last season's Championship winners, offered some threat on the break themselves. On 17 minutes, a dangerous cross from Ritchie De Laet on the overlap down the right had to be headed behind by Ivanovic.
Algerian Riyad Mahrez forced a decent save from Courtois with a low strike after weaving into space at the edge of the Chelsea penalty area.
Chelsea looked to step up a gear as Costa pulled the trigger from 18 yards, but Foxes captain Morgan flung himself in the way to deflect the ball over.
Blues boss Jose Mourinho was out of his technical area claiming a penalty moments later when Costa went down under a strong challenge from the defender, but referee Lee Mason was unmoved, which on replays looked the correct call.
Oscar was unfortunate not to give Chelsea the lead at the start of the second half when his curling 20-yard effort came back off the inside of the far post. Ivanovic's header was palmed away by Schmeichel from the follow-up attack.
Schurrle was then flattened by a sliding challenge from Dean Hammond on the far touchline - which had the Chelsea bench remonstrating with the fourth official, but referee Mason produced only a yellow card.
Leicester closed down the free-kick and broke quickly, with the ball eventually worked out to Nugent on the left side of the Chelsea penalty area. His low shot was deflcted goalwards and kicked away by Courtois.
As the match opened up, Ivanovic was denied by another fine save from the Leicester keeper, this time an angled drive touched over.
The visitors could have taken the lead on 55 minutes when Nugent beat the offside trap to run clear on goal, but Courtois made a smart save to push the ball wide.
Almost straight away and Chelsea were in at the other end, as Fabrgeas ghosted into the right side of the Leicester penalty area, where his angled chip was palmed behind by Schmeichel.
The breakthrough finally came on 63 minutes. Oscar played Ivanovic into the right side of the area, and his cutback was chested down by Costa, before he drilled the ball past Schmeichel from six yards.
Chelsea immediately made a change as Schurrle was replaced by Ramires.
On 68 minutes, Leicester sent on Marc Albrighton to make his debut after joining from Aston Villa, with Mahrez coming off. Hammond was then also substituted, hobbling into the away dugout as Gary Taylor-Fletcher came on.
Chelsea looked to have wrapped things up with 13 minutes left when Hazard collected the ball on the left, cut inside before drilling a low shot past Schmeichel into the corner.
Stamford Bridge was soon rocking again when Didier Drogba was sent on for a cameo, the veteran striker having rejoined the club in the summer - with owner Roman Abramovich leading the applause.
With two minutes left, Schmeichel tipped a 25-yard drive from substitute Willian over the crossbar.
Chelsea, who stay top on goal difference from Swansea after two games played, travel to Everton next Saturday, while Leicester - who are 14th in the embryonic table - host Shrewsbury in the Capital One Cup second round on Tuesday and then Arsenal at the King Power Stadium in the league the following Sunday.
Matt Le Tissier on Chelsea v Leicester
I was really impressed with Leicester, in the first half especially. They set up really well, they countered well and quick and when they went forward, they threw numbers at Chelsea. I tipped them to struggle and perhaps get relegated, but I’ve watched them in the first two games and I’d be very surprised if that team went down.
I don’t know what Jose Mourinho said to Chelsea at half-time, but they came out firing on all cylinders, especially Ivanovic who was their best attacking force. He was up and down that touchline and in the box at every possible opportunity.