Making his first league start for the Blues, Ziyech scored and assisted as Chelsea eased to a 3-0 win at Turf Moor.
Also on the scoresheet was defender Kurt Zouma and striker Timo Werner, who was a late replacement for the injured Christian Pulisic.
Speaking after Chelsea’s win, Ziyech said: ‘It’s been a good week for me. Hopefully we can continue. I feel good and feel at home. Everything is fine.
‘Especially after European games it’s always difficult. I think we had a good game, started well and didn’t really have problems at all.
‘Burnley had one chance at the beginning, but after that we controlled the game.
‘You need to play with confidence and not be afraid to make mistakes. Otherwise it won’t go how it’s supposed to go.
‘Every week we make steps. Let’s keep it that way.’
Ziyech’s Chelsea team-mate Mason Mount, meanwhile, added: ‘We know how hard it is to come here and get a result.
‘Brilliant to get three goals and get a clean sheet as well.
‘It’s always going to take some time for us to click as a team but today it felt like we was moving with each other, we knew where each other wanted to go, and it clicked today.
‘It’s exciting stuff when you’re playing like that – now we need to look at keeping that going.
‘There’s a lot of competition in the squad. We’re all working hard, we’re like a family. All we want to do is win as many games as we can.
‘Getting that second goal from Big Zoum, that kind of set piece we didn’t really have last season and it’s something we’ve done a few times already this season.’