Vincent Kompany's men enjoyed the majority of the ball but could not find a way past Jordan Pickford in the visitors' net, condemning the Clarets to their eighth loss in nine games at their Lancashire home since returning to the top flight.
The Toffees went ahead early in the evening thanks to a powerful header from Amadou Onana, before ex-Claret Michael Keane added a second to send the Merseysiders well on their way to victory.
Following the 1-1 draw at Brighton & Hove Albion last weekend, Burnley were forced into two changes to their starting XI, with Hannes Delcroix entering the fray due to Charlie Taylor's suspension, and Jacob Bruun Larsen replacing Johann Berg Gudmundsson, who did not make the Clarets' matchday squad.
One-game bans for Jarrad Branthwaite and Idrissa Gueye allowed Onana and Keane to start the contest in Lancashire, whilst Jamie Patterson and Ben Godfrey formed part of the visitors' back-five formation as Ashley Young and Vitalii Mykolenko were ruled out due to their respective knocks.
With Sean Dyche's Toffees opting for a defensive-minded approach in terms of their formation, it was no surprise to anyone inside Turf Moor to see Burnley dominate possession in the opening stages, however the physical presence of Dominic Calvert-Lewin in the Toffees' attack always provided them with an out ball.
Any hopes of the Clarets recording back-to-back clean sheets at home in the Premier League were wiped away after just 19 minutes of the match, when Dyche's men made their aerial dominance count as they got well on their way to a fourth-straight top-flight victory since their 10-point deduction.
Former Burnley man Dwight McNeil whipped in a tempting delivery from a corner towards the back post, and Onana was on hand to emphatically head the ball into the back of James Trafford's net, after the Belgian had easily evaded the attentions of marker Josh Brownhill.
It was not long before Everton doubled their advantage in Lancashire, with the visitors once again making use of Burnley's frailties in the air, and this time it was former Claret defender Keane who got the better of £19m summer arrival Trafford in between the hosts' sticks.
England's Pickford sent a long free kick towards the Burnley box, where James Tarkowski won the first ball and directed his touch towards centre-back partner Keane, who - after seeing his initial effort saved by Trafford and rebound back - slotted his second attempt into the near post.
Kompany's side probed at a stubborn Everton backline following the visitors' second and could have reduced the deficit to one goal in stoppage time of the first half, however a sliding clearance by Godfrey denied Zeki Amdouni a tap-in following Vitinho's low cross from the right.
More to follow.