The 27-year-old slotted Chelsea’s winner one minute from time, but could so easily have been sent off for a ragged second-half challenge on West Ham’s goalkeeper.
Costa was booked for dissent in a goalless first-half, but referee Anthony Taylor refused to take action when the Spain forward upended Adrian after the interval.
Eden Hazard’s well-struck penalty handed Chelsea the early second-half lead, only for defender James Collins to produce a fine equalising strike.
But just when West Ham thought they would hold firm for a hard-fought point, up popped Costa with a smartly-struck winner, but one that would doubtless anger visiting boss Slaven Bilic.
Former Juventus manager Conte revealed he admires gritty striker Costa’s fiery approach in midweek, but the ex-Italy boss may well feel the 27-year-old overstepped the mark with his rash challenge on Adrian.
All that fuss had no impact on the scoreboard however, with ex-Italy boss Conte able to get his Premier League campaign off to a winning, albeit breathless start.
A cagey first-half yielded just two saves in total, West Ham stopper Adrian holding firm for both Branislav Ivanovic’s near-post nudge and Willian’s curling free-kick.
Hazard ought to have handed Chelsea the lead after ghosting into West Ham’s box on a scything left-wing foray.
The Belgium forward curled his low effort wide of the upright however, leaving the animated Conte to slap his thigh in frustration on the touchline.
Oscar had a spurious penalty appeal waved away after light contact from Winston Reid, with Costa booked for dissent in the obligatory inquest.
West Ham certainly missed Dimitri Payet’s ingenuity from the outset but the France star started on the bench given his phased reintegration after Euro2016.
All Chelsea’s star names were on show by contrast, and one leading light who dimmed markedly last season stepped up to hand the hosts the lead.
Hazard blasted gleefully into the roof of the net from the penalty spot to open the second-half, after Michail Antonio had fouled Cesar Azpilicueta.
Belgium star Hazard needed 27 games to get off the Premier League mark in a torrid battle for form last season but suffered no confidence crisis as he blasted the hosts into the lead.
West Ham boss Bilic made his feelings on Antonio’s crude challenge on Azpilicueta clear for all to see, by hooking the defender just moments after Chelsea took the lead.
Willian forced a fine save from Adrian with a neat shot across goal as Chelsea sought to ram home their dominance, leaving Conte kicking at thin air in frustration on the touchline.
Payet joined the fray with some 25 minutes on the clock, paving the way to help haul the visitors back into the contest.
Just when Chelsea sought to close out an untroubled victory though, Collins turned the unlikely hero.
There was nothing rough about the rugged defender’s crip equalising strike, the centre-back blasting home with 15 minutes to play.
The leveller had West Ham boss Bilic jumping for joy on the touchline, but that elation proved all too short-lived thanks to Costa’s late winner.