Despite the Clarets starting on the front foot at Turf Moor and pushing for a way back into the encounter in the second half, two first half goals from the Toffees proved to be the difference on Saturday evening.
Andre Onana gave Everton the lead from a corner before former Claret Michael Keane turned home from close range to double the advantage.
Manager Vincent Kompany made two changes from the side that drew at Brighton last weekend as Jacob Bruun Larsen and Hannes Delcroix came in for Johann Berg Gudmundsson and the suspended Charlie Taylor.
Forward Lyle Foster was named amongst the substitutes and involved in the matchday squad for the first time since taking a break from playing in October before coming on at half time. Manuel Benson was also used as a second half substitute after not featuring since September.
First Half
After a cagey start at Turf Moor, the Blues carved open the first opportunity of the game with nine minutes played when Harrison curled a teasing ball towards the back post for McNeil to guide inches wide of the post with a header.
Everton had another chance shortly after when Patterson picked out Calvert-Lewin a few yards out but the forward saw his header denied well by Trafford and pushed wide for a corner.
But from the following corner, the Toffees broke the deadlock as Onana rose highest and powered home a header from close range to give the visitors the lead.
The Blues then had a second of the night moments later when Keane’s rebounded strike fell back into the path of former Burnley defender to slot in from yards out.
As the Clarets looked for a way back into it with the break approaching, Vitinho’s fizzing low ball into the area aiming for Amdouni was intercepted at the last second by Godfrey to deny a clear goal scoring opportunity for the hosts.
Second Half
As the second half restarted, Kompany made a half-time change with Foster replacing Rodriguez in attack for his first appearance since Brentford.
It was the Clarets who had the first chance of the second half when Vitinho slipped in Bruun Larsen but the Dane forward could only fire into the hands of Pickford.
Pickford was then forced into another save, this time having to tip it wide for a corner, as Amdouni’s stinging effort was palmed away by the England goalkeeper.
The Clarets came close to pulling a goal back with 10 minutes left on the clock when Berge rifled a shot off the inside of the crossbar from outside the area.
All the action from Turf Moor
Burnley: Trafford, Vitinho (Redmond 87), O’Shea, Beyer, Delcroix (Al-Dakhil 68), Berge, Brownhill, Amdouni, Bruun Larsen (Benson 81), Odobert (Tresor 87), Rodriguez (Foster HT)
Unused substitutes: Muric, Roberts, Cullen, Obafemi,
Everton: Pickford, Patterson, Godfrey, Keane, Tarkowski, McNeil, Garner, Onana, Doucoure (Dobbin HT), Harrison, Calvert-Lewin (Beto 81)
Unused substitutes: Virginia, Lonergan, Hunt, Campbell, Metcalfe, Chermiti, Danjuma
Goals: Onana 19, Keane 25’