Pedro Neto struck shortly after half-time and added a second in stoppage time to end Everton's unbeaten start to the Premier League season. The defeat also brought concern over Chelsea goalkeeper Emiliano Martínez, who was stretchered off following a clash of heads with Jarrad Branthwaite.
Moyes was particularly unhappy that Everton midfielder Harrison Armstrong had been booked for a challenge on Josh Acheampong, believing the tackle was not deserving of a caution.
“If our football in this country is not going to allow them then I don’t know. We’ve got referees who we want to be the best in Europe, or think they’re the best, because they let things go,” he said.
> “I thought it looked a half-decent tackle. Maybe somebody might tell me I’m wrong or tell me differently, but I wasn’t sure that it was the right decision.”
The Everton boss also believed Chelsea forward Morgan Rogers could have received a second yellow card for his challenge on James Garner.
“I had a wee look at the Morgan Rogers tackle on Jimmy, which, maybe some might have said something different, and I think that might have made a second yellow, it wasn’t the case,” Moyes added.
Despite his complaints, Moyes accepted that Chelsea had been the better side overall, although he felt the match had been competitive for long spells. He was especially disappointed with the opening goal, when a deflected cutback fell kindly for Neto, and felt the second goal also involved an element of fortune after a free-kick bounced into Chelsea's path.
Xabi Alonso, meanwhile, saw the performance as evidence that Chelsea are learning to manage difficult away fixtures. Speaking to the club's official website, he said: “We are very happy with the result and the performance. We needed one of these wins away, where we have played very professionally and seriously, looking solid. It was a very good win. We got many things right.”
Alonso praised the team's composure after a fractious first half and welcomed a clean sheet, Chelsea's first in the league this season.
“We looked in control of the game, there were moments for sure when we had to defend, especially in the first half, but overall the team looked mature, knew what to do in each moment, and knew we’d have our own moment,” he said.
For Moyes, the task now is to turn competitive spells into results, while Chelsea will hope their disciplined display can become a foundation for greater consistency.