Ismaila Sarr gave the visitors the lead after only 12 minutes and, while the deficit remained at just one going into the interval, it could have been worse with the Hornets also hitting the post twice.
There were no such let-offs for Stoke in the second half, though, as Ken Sema, Keinan Davis and Vakoun Bayo all scored in the final half an hour to put Alex Neil's side to the sword.
It was a dream first game back in English football for Bilic, who in the buildup to the match had played down concerns over the Watford owners' ruthless record of sacking managers, having become the club's 10th permanent boss in the past five years.
Bilic's predecessor Rob Edwards was given just 11 games before he was sacked with Watford having won just one of their last eight, but performances like today's should ensure that the new manager has longer at the helm.
Sarr set the Hornets on their way to their biggest win since February 2021, narrowly springing the offside trap to nod home Hassane Kamara's fine cross from the left.
Sarr and Sema then hit the same post within two minutes of each other as Watford pushed for more, with Liam Delap offering the only slight glimpses of a response at the other end for Stoke.
Sema did get his name on the scoresheet when he scrambled home from close range shortly after the hour mark, while another slack piece of defensive work contributed to Davis grabbing a third.
Bayo capped off the scoring when he poked home Imran Louza's dangerous cross at the back post, prompting the home fans to boo their own side off the field at full time.
Alex Neil - himself relatively new in the job - must now pick his side up from the defeat ahead of the midweek trip to Burnley, with Stoke staying just two points clear of the relegation zone courtesy of this result.
Watford, meanwhile, host Swansea City on Wednesday looking to record back-to-back wins for the first time since their opening two games of the campaign.