The reaction of Roman Abramovich –-- marking the 11th anniversary of his first home game as club owner –- summed up the mood in Stamford Bridge, with the Russian standing to applaud Costa when he was later replaced by Didier Drogba.
After the failure of a number of high-profile expensive strikers at the London side, most notably Fernando Torres, Costa's start to life in English football following his £32 million ($55 million, 40 million euros) arrival from Atletico Madrid has given cause for optimism.
While Costa impressed, a further question mark was placed against fellow Spain international Torres's future after he was omitted from the match-day squad.
Not that this was an entirely comfortable win for Portuguese manager Mourinho's Chelsea side.
But ultimately the outcome of the game was not in doubt and the margin of victory could have been greater but for the efforts of City keeper Kasper Schmeichel.
Chelsea had demonstrated their title credentials in securing a comfortable three points at Burnley, rallying impressively after falling behind to an early goal.
Leicester, though, quickly showed they would present a more formidable obstacle than their fellow newly-promoted team, with the visitors remaining largely untroubled during the opening 45 minutes.
The slick interplay that had marked the win over Burnley was missing and an early mix up between Costa and Andre Schurrle summed up Chelsea's disjointed start.
Costa, though, came close to testing Schmeichel with a powerful, 13th-minute drive.
That was one of the few occasions Chelsea threatened during the first half, with Leicester looking increasingly comfortable as the interval approached.
And had Riyad Mahrez, the Algeria winger, shown more composure on the three occasions he found himself clear on the right of the Chelsea penalty area, Nigel Pearson's side might have ended the half in front.
- Agitated Mourinho -
Mourinho had been growing increasingly agitated on the touchline and his side's approach after the re-start suggested the manager had managed to get his message across at half-time.
Chelsea immediately pressed forward and Oscar came within inches of making the breakthrough in the 48th minute when he curled a low, angled shot against the inside of the post.
Schmeichel, son of Manchester United goalkeeping great Peter, then produced the first of a succession of fine saves to block Branislav Ivanovic’s header after the full-back connected with Cesar Azpilicueta's cross.
The keeper would produce another full-length effort to keep out Ivanovic's long-range shot and an excellent reaction save to tip over Cesc Fabregas's attempted chip.
But the best chance of the opening hour fell to City who should have taken a 55th minute lead when Mahrez sent David Nugent clear 40 yards from goal only for the forward to shoot too close to Courtois, allowing the keeper to save with his legs.
There was a sense Leicester would not get a better chance, and Nugent’s miss appeared even more costly when Costa put Chelsea ahead in the 63rd minute.
Ivanovic had become increasingly influential with his attacking runs from right-back and the defender again worked his way into the Leicester box to collect a fine pass from Oscar.
The Serb delivered a low cross that was deflected towards Costa off Liam Moore and the forward controlled on his chest before finishing with a close range, right-foot shot.
That strike sapped the visitors self-belief and the result was out beyond doubt when Hazard, cutting in from the left, produced an excellent finish for Chelsea's second goal.