Michy Batshuayi's goal eight minutes from time earned Conte's side a 1-0 win at West Brom and ensured they clinched the trophy at the earliest opportunity
The Italian had already been drenched by champagne, iced water and beer in the dressing room, forcing a change of clothes, before his post-match media conference was brought to an abrupt halt by Costa and David Luiz.
The pair then impatiently waited, with club captain John Terry standing guard to make sure nothing untoward happened, as Conte fulfilled a further media commitment.
And Costa held a fire extinguisher aloft in the direction of journalists as he attempted to speed up proceedings and resume celebrations with the head coach who had steered Chelsea to a second title in three years after last season's 10th-placed finish.
Conte, who has won the last four domestic championships in which he has managed having claimed three in a row with Juventus prior to being Italy boss, could win the double in his first campaign in England if the Blues can beat Arsenal in the May 27 FA Cup final.
"It's amazing," Conte told Chelsea TV
"I want to say thanks to my players, my staff, the club, who always support me.
"Now it's important to rest and we want to try to win also the FA Cup and for this great season to become a fantastic season.
"I'm living great emotion at this moment."
Conte enjoyed the on-pitch celebrations with the Chelsea supporters.
He added: "It's great to see this link with the players, with our fans
We have to stay always together, to win, to lose together, but always to fight."
Conte saw the potential in Chelsea, but it took time to turn his players into title challengers once more.
At his unveiling on July 14 last year, Conte said: "I hope there is a small flame flickering that can eventually grow into a blazing inferno."
Costa had threatened to douse the flames and jeopardise the title bid in January, when he apparently had his head turned by a lucrative offer to join Chinese Premier League side Tianjin Quanjian.
Conte refused to discuss reports of a disagreement with Costa in public, attributing the striker's absence from the Premier League trip to Leicester to a back injury
Yet Costa's future is far from certain, with rumours recently resurfacing of a possible move away from Stamford Bridge this summer.
Costa has scored 20 Premier League goals this term and the top flight's top-scorer, Everton's Romelu Lukaku, has been linked with a return to Chelsea.
Lukaku, who has scored 24 times this season, added to speculation by commenting "Aaah michy" and four laughing emojis as Belgium team-mate Thibaut Courtois shared live video on social media of the dressing-room celebrations.
Source : PA
Source: PA