Morata's three-goal feat was Chelsea's first since Diego Costa bagged a hat-trick against Swansea at the start of the 2014/15 campaign.
Micky Batshuayi scored a midweek hat-trick against the Championship side but starts on the bench today.
Stoke missed a hatful of good chances in the second-half, especially through Fletcher and Choupo-Moting, and were punished for their lack of efficiency through Morata's late double - which completed his hat-trick.
"I am pleased for him and the performance of the team".
The win extends Stoke's wait for a victory to five matches across all competitions, whereas Chelsea are now unbeaten in seven games and have won their opening three away fixtures of a league season for the first time since 2009. "This game is very hard - I remember last season".
"We are talking two players who are very strong, two really good players, two number nines who are very strong".
But it also means Morata is the undisputed No. 9 at Stamford Bridge and the club's success, or otherwise, will largely be down to how many goals he scores.
Chelsea's head of communications and public affairs Steve Atkins added: " We spoke to Alvaro after the game and he is grateful also that they have changed the words".
Antonio Conte guided Chelsea to a 4-0 win over Stoke.
"I spoke to the doctor before the third and final substitution and I asked if he was OK", Conte said of Courtois. The referee allowed them to stop the game, take away all momentum and he made a call with Marcos Alonso who in my opinion could have been sent off. "I think the last two goals embellished the result".
Jese, who scored an impressive debut victor against Arsenal last month, was anonymous against Chelsea and it wasn't until Peter Crouch entered the action midway through the second half that Stoke began to hurt the champions.
Conte's stand-off with Costa had proved an unwelcome distraction this season after the Chelsea forward was told by the Italian that he was no longer part of his plans. "Our fans were fantastic, they pushed us a lot".