Costa, who had opened his account in the 3-1 victory at Burnley on Monday night, was on hand to convert from close range just after the hour, which ended the brave resistance of the Foxes.
Eden Hazard stroked in a second for Jose Mourinho's men on 77 minutes.
Leciester, 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 championship once again.
A deflected shot from Brazil midfielder Oscar looped up over Kasper 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 Branislav 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 Wes Morgan flung himself in the way to deflect the ball over.
Blues boss 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 Cesc 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 the Brazil-born Spain international, signed from Atletico Madrid, drilled it past Schmeichel from six yards.
Chelsea immediately made a change as Schurrle was replaced by Ramires.
On 68 minutes, Leciester 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.