The Ivorian capitalised upon two poor errors by Jordan Pickford and Ben Godfrey in the 85th minute to earn Sean Dyche's side only their fourth league win of the season.
It lifted them to within one point of safety behind Frank Lampard's fourth-bottom side.
In an entertaining and feisty encounter, the hosts took a 12th-minute lead when an unmarked Nathan Collins scored at the back post from a corner.
But the advantage did not last long as Richarlison stroked home the first of two penalties in the 18th minute when Ashley Westwood was adjudged to have tripped Anthony Gordon.
His second came four minutes before half-time and was tinged with controversy.
Despite Vitaliy Mykolenko being tripped by Aaron Lennon, referee Mike Dean did not think it amounted to a spot-kick, until he was advised by the video assistant referee to review it on the monitor and then overturned his decision.
The home support showed their disgust with that call, but their mood improved after the break when a great burst by Charlie Taylor led to him crossing for Jay Rodriguez's equaliser in the 57th minute.
And although Everton created the better chances as the second half wore on, Cornet's seventh goal of the season could not have been more crucial as it sent the home fans wild.
By contrast, the Everton players and boss Lampard left the field dejected, with their 68-year stay in the top flight under serious pressure.
The victory was as morale-boosting for Burnley as it was devastating for Everton, who have earned only two wins from nine league games since Lampard replaced Rafael Benitez in January.
With wet and blustery conditions, this had all the ingredients for a relegation scrap, something Burnley have had more experience of in recent years compared to the visitors.
But Everton matched their opponents for much of the game, and were on top for most of the second half as they sought to earn a success that would boost their own survival hopes.
Richarlison almost had a hat-trick when his overhead kick was saved by Nick Pope.
Dominic Calvert-Lewin then headed over, while Gordon had a great opportunity blocked by James Tarkowski when clean through.
However, as had been the case for much of the season, Everton were undone by individual errors as their wretched away form continued.
Pickford mis-kicked a clearance out of play with his right foot, which led to the throw-in from which Burnley got their winner. It was made worse when defender Godfrey also sliced a clearance from Matej Vydra's cross, allowing Cornet to finish.
Everton now face a daunting task to stay in the league - and relegation would cause huge problems for a team that has registered heavy losses over the past three years.
