A goal before the break looked very unlikely at first, with neither side offering any threat going forward in the opening 30 minutes.
However, just after the half-hour mark, the Toffees managed to get their noses in front.
A free-kick into the area from James Garner found James Tarkowski unmarked, with the defender having space to run onto the ball and head it past a committed Martin Dúbravka to open the scoring.
Burnley appeared to still be feeling the negative effects of their demoralising 4-3 defeat at home to Brentford on Saturday, in a game which saw them battle back from 3-0 down only to lose out in stoppage time.
The Clarets only mustered a solitary shot in the first half, leaving themselves with work to do once more, as they trailed at half time for a league-high 16th time in the PL this season.
With Everton very much in command in terms of general play, they were keen to find a second goal to ease any potential nerves inside the final half-hour. And on the hour mark, they managed to do just that with their first shot on target of the second half.
The ball was worked to Iliman Ndiaye just outside the area, and the Senegalese international slipped it through to fellow midfielder Kiernan Dewsbury-Hall, who lifted his effort from inside the area over Dúbravka to double the home side’s advantage.
Perhaps still harbouring memories of recent capitulations at the Hill-Dickinson Stadium, Everton played it safe for the remainder, meaning that the Toffees' vital second goal proved to be the final moment of note on a truly miserable evening for the Clarets and their long-suffering supporters.
In response to that, the visitors only managed to fashion two shots on target in the second half on the way to an 18th defeat of the season.
As well as emerging victorious at the Hill Dickinson Stadium for the first time in three months, Everton also ended their four-game winless run at home on Tuesday.
They remain eighth and still very much in touch with the sides around them, chasing a European place.
Burnley, meanwhile, seem destined for a third relegation from the PL to the Championship in the last five seasons – they stay 19th and eight points adrift of safety with just nine matches remaining.
Flashscore Man of the Match: James Garner (Everton)