The Toffees named the same starting XI who began the goalless draw with Crystal Palace last time out, while the visitors recalled Robbie Brady and Johann Berg Gudmundsson, with Jeff Hendrick and Dwight McNeil dropping to the bench.
The hosts made a fast start, forcing several corners as the Clarets were penned back deep in defence.
Richarlison curled a 20-yard effort over the crossbar ten minutes in as the home dominance continued and it was the Brazilian forward who was influential in the deadlock being broken with 17 minutes on the clock.
Gylfi Sigurdsson laid the ball off to Richarlison 25 yards from goal and his left-foot effort took a crucial deflection off Ben Mee to leave Tom Heaton well beaten.
Three minutes later it was 2-0, as Heaton could only parry a powerful shot from distance from Lucas Digne and the alert, unmarked Seamus Coleman pounced to head home the loose ball from close range.
Ashley Barnes had the ball in the net for the visitors just past the half-hour mark, but an offside flag was up to cut short the celebrations before they had even started.
Early in the second half Richarlison was forced off due to a knock, with Theo Walcott taking the place of the South American.
The visitors threatened for once as Barnes headed the ball down to strike partner Chris Wood, but his mishit shot was far too weak to really extend Jordan Pickford.
Dominic Calvert-Lewin rifled a shot over the top from just outside the area as the hosts pushed forward once more, before the Clarets brought on both McNeil and Matej Vydra to try and find a way back into the game.
Fifteen minutes from time Vydra appeared to be bundled over in the area by the challenge of Michael Keane, but referee Kavanagh waived away the appeals from the visitors.
And that proved to be the last threat of note for the Clarets, with the Toffees almost adding a third goal late on as sub Ademola Lookman lashed in a shot which flicked the top of the crossbar.