Morata struck three times at the bet365 Stadium yesterday to take his Chelsea tally to seven goals in six games since his close-season move from Atletico's local rivals Real Madrid.
There was also relief, for Morata and manager Antonio Conte, that Chelsea's fans dropped a controversial chant about the Spain striker that contained an anti-Semitic reference to rivals Tottenham Hotspur.
"Our fans were fantastic today and they pushed a lot the team in every moment of the game," said Conte, whose side follow their trip to Spain with a home game against Manchester City next weekend.
"I think Alvaro performed very well. He scored a hat- trick and for our strikers it's very important.
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"It means we created chances for our strikers to take, but I am pleased for his performance and I'm pleased for the performance of the team."
Morata had previously asked Chelsea's fans not to sing the song about him and they responded by dreaming up a different chant in his honour.
- Courtois unscathed -
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"It's not easy to win away against Stoke. This game is always difficult," added Conte, whose side trail leaders Manchester City and Manchester United by three points.
"I remember very well last season when we struggled a lot to win and we also won at the end of the game.
"Today we started very well and then we started to suffer and it was important not to lose our composure, to always stay in the game.
Conte confirmed goalkeeper Thibaut Courtois had suffered no ill-effects from a heavy landing on his neck during the second half and will be fit to face his former club Atletico.
Stoke manager Mark Hughes had no such comfort after he saw his side undone by horrible defensive errors, then reported his centre-back crisis could extend into next week's home game against Southampton.
Hughes will have Kurt Zouma available after the Chelsea defender was prevented from playing against Conte's men by the terms of his loan to Stoke.
Hughes said: "When you go up against a side with the quality of Chelsea, you've got to make sure you don't have catastrophic defensive moments and we had at least three, so that's always going to make it difficult.
"There were extenuating circumstances with our difficulties with personnel at the back and with Bruno coming off.
"We've got Kurt back next week, but all the others are out so we could struggle again next week.
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