Morata, the club-record signing bought to replace Costa, was on the scoresheet again early on before Pedro extended the lead on the half-hour mark.
Both goals came from defensive mistakes by Stoke, whose manager Mark Hughes had demanded a response from his players following the disappointing midweek defeat at Bristol City in the Carabao Cup. He did not get it, with Morata rounding off the win in the final quarter-hour.
"Alvaro performed very well," praised Conte. "He scored a hat-trick and for a striker this is very important, and don't forget the last game it happened the same with Batshuayi. It means we create chances for our strikers.
"I am pleased for his performance and the performance of the team. It is not easy to win away games against Stoke, this game is always difficult. I remember very well last season when we struggled a lot to win and we won at the end of the game.
"Today we started very well and then we were good when suffering [pressure from Stoke]. It was important not to lose our composure and to stay always in the game, and then we scored two goals and we created many chances to improve the score. It was 4-0 but it was a difficult game."