Manager Sean Dyche enjoyed a winning return to Burnley as his in-form Everton side picked up their fourth consecutive Premier League victory.
Dyche was in charge at Turf Moor for almost 10 years before being sacked last year, and there was no room for sentiment as his Toffees team ruthlessly moved seven points clear of the relegation zone.
James Trafford made a superb save to keep out Dominic Calvert-Lewin's header, but Amadou Onana leapt above the Burnley goalkeeper from the resulting corner to nod in the opener from Dwight McNeil's delivery.
The visitors doubled their lead after 25 minutes as Michael Keane stroked in at the second attempt after his initial effort was saved by Trafford and Dara O'Shea failed to clear the rebound.