Ashley Barnes had Burnley's best chance on five minutes but, after controlling a ball over the top, the striker shot over.
Ziyech's first Premier League goal came in the 26th minute, when the Moroccan converted a low shot that wrongfooted Nick Pope.
Chelsea doubled their lead on 63 minutes when Mason Mount’s corner was powerfully headed in by Kurt Zouma.
Ziyech turned provider seven minutes later for Timo Werner to make it 3-0, with the German only starting after Christian Pulisic was injured in the warm-up.
Chelsea are five unbeaten and rise to fourth with 12 points.
Burnley are bottom with just one point.
Budweiser King of the Match: Hakim Ziyech (CHE)
Burnley: 6 Nov v Brighton (A)
Chelsea: 7 Nov v Sheff Utd (H)