Premier League: Chelsea 2 Leicester 0
Jose Mourinho's title hopefuls turned in a dazzling display to beat Burnley 3-1 in their opening fixture, with Spaniard Costa starring, but they found another newly promoted team a far tougher nut to crack on Saturday.
Home goalkeeper Thibaut Courtois was every bit as busy as his Leicester counterpart Kasper Schmeichel during the opening hour, with David Nugent’s failure to capitalise on a 55th-minute opening particularly costly for the visitors.
But, following a low-key first half, Mourinho's decision to push his full-backs forward after the interval paid dividends, with Branislav Ivanovic creating a scruffy 63rd-minute goal for Costa to break the deadlock.
It continued the flying start to his Chelsea career for the big-money arrival from Atletico Madrid and provided Mourinho with a welcome display of the cutting edge that Chelsea lacked in their push for Premier League honours last season.
Eden Hazard once again showed his class with a wonderful solo effort to seal the points, netting the second with 13 minutes left.
The hosts pushed forward early on and Cesc Fabregas - who had performed a star turn on his debut at Burnley - played in Costa to set up Andre Schurrle in the eighth minute, but the Germany international shot over.
Gary Cahill then had a shot blocked when Schmeichel flapped at a 20th-minute corner.
Nevertheless, the visitors made a confident start as Nugent, Jeffrey Schlupp and Riyad Mahrez all asked questions of the Chelsea defence.
Schurrle headed over when a previously low-key Hazard shimmied his way to the byline, before Mahrez's own nifty footwork left Cesar Azpilicueta in his wake and created enough room for the winger to fire in a rasping 32nd-minute drive that Courtois had to beat away to safety.
The impressive Mahrez was guilty of overplaying as Azpilicueta snuffed out another promising attack and Chelsea responded, with City centre-back Wes Morgan flinging himself forward to divert Costa's 36th-minute strike over the crossbar.
Oscar clipped an 18-yard effort against the the inside of post three minutes after half-time and, as Chelsea kept the ball alive, Ivanovic headed goalwards from Azpilicueta's cross to draw a fine stop from Schmeichel.
Courtois then found himself back in action to keep out Nugent's deflected attempt on the counter-attack, although the sight of Schurrle curling a shot over the crossbar underlined Chelsea's continued threat.
Oscar had a 54th-minute penalty claim for handball rejected and, a few seconds later, Ivanovic hammered in a shot from distance to extend Schmeichel once more.
Again, though, Courtois was forced to outdo his opposite number, saving superbly after Nugent had raced clear one-on-one only to produce a shot that lacked conviction.
Schmeichel still appeared destined to be the busier of the two, clawing Fabregas' dinked chip behind in the 58th minute before Leicester's resistance was finally broken five minutes later.
Oscar sent Ivanovic racing into the City box and he found Costa, who scrambled the ball home under pressure.
Leicester faded as an attacking force thereafter while Chelsea introduced Didier Drogba late on for his second home Premier League bow.
But, despite seeing Hazard jink beyond substitute Marc Albrighton to rifle in a 77th-minute finish, Leicester boss Nigel Pearson was given plenty of reason to be optimistic for the challenges ahead.
Chelsea, meanwhile, already look every bit title challengers.