Injuries delayed Ziyech's introduction to the Chelsea team, but he followed up his goal in the 4-0 Champions League win over Krasnodar in midweek by registering the opener at Turf Moor.
Kurt Zouma had a couple of sloppy moments in the first half but there was no doubt about the powerful header he produced to double the visitors' lead in the 63rd minute.
Ziyech then teed up Timo Werner – who only started due to an injury Christian Pulisic sustained in the warm-up – to hand Burnley a fourth straight home Premier League defeat for the first time since April 2010.
Ashley Barnes collided with Edouard Mendy after lifting a shot over the goalkeeper and the crossbar in the fifth minute, though referee David Coote did not deem it worthy of a penalty.
Nor did the official point to the spot when Ben Chilwell was nudged out from under Reece James' cross by Josh Brownhill, with play continuing and Nick Pope having to tip a looping Tammy Abraham header behind.
After Zouma stopped Barnes punishing his error near the halfway line, Pope was caught out when Ziyech fired a low drive from the edge of the box through Dale Stephens' legs and into the bottom-right corner in the 26th minute.
James Tarkowski made an important intervention to deny Werner a tap-in before half-time, but he was lost at sea when Zouma rose to steer Mason Mount's corner into the net.
Burnley unravelled and after Ashley Westwood's cross-field ball was intercepted by James, Ziyech slid a pass through for Werner to put the game to bed in the 70th minute.
What does it mean? Lampard making strides with defence
Their shutout at Old Trafford last weekend was just their second on their travels in the Premier League under Frank Lampard – they have now achieved back-to-back clean sheets on the road in the top flight for the first time since October 2018.
Mendy replacing Kepa Arrizabalaga has helped tighten things up, with the former Rennes keeper emulating club great Petr Cech, who, in August 2004, was the last Blues keeper to avoid conceding in each of his first three Premier League appearances.
Hakim the dream
Ziyech became the first Chelsea player since Diego Costa in 2014 to score in both of his first two starts in all competitions, but he also provided a display of the range of passing that makes him such a threat from the wing.
Burnley delivered nine crosses from open play but failed to register a single shot on target. Of the five attempts they did muster, only one of them came from inside the box. Sean Dyche's side lack a cohesive plan going forward and a trip to Turf Moor no longer represents the tough challenge it used to.
Key Opta Facts
- Chelsea have won six of their seven away Premier League matches against Burnley (D1), winning each of their last four in a row.
- Burnley have picked up just one point in their opening six league games this season, their worst ever start to a league season in their history.
- Edouard Mendy has conceded just one goal in six appearances for Chelsea in all competitions – Kepa Arrizabalaga has conceded 14 goals in his past six appearances for the Blues.
- Since the start of last season, Ziyech has been directly involved in 30 goals in 40 matches in all competitions across spells with Ajax and Chelsea (11 goals, 19 assists).
- Ziyech is the first player to score and assist on his first Premier League start since Ruben Neves in August 2018, and the third African player to ever do so, after Mohamed Salah and Dame N'Doye.
Chelsea are back in Champions League action at home to Rennes on Wednesday, while Burnley have a Premier League trip to Brighton and Hove Albion on Friday to look forward to.