The victory saw Burnley remain in the relegation zone but move to within a point of Everton.
Burnley took an early lead through Nathan Collins' first goal for the club but fell behind before halftime as Richarlison twice scored from the penalty spot.
Jay Rodriguez equalised in the 56th minute before Cornet's first goal since his return from the Africa Cup of Nations in January gave Burnley their first win in six games.
It was also just the Clarets' fourth win in 29 league games so far this season, with Burnley having lost their last four games without scoring a goal.
The result meant Everton had now suffered six straight away defeats in the Premier League and facing the prospect of playing second-tier football for the first time in 68 years, with manager Frank Lampard's dismay visibly evident at fulltime.
Collins broke the deadlock with 12 minutes gone after yet another defensive error by Everton meant he hooked in a Cornet corner that had failed to be cleared by several defenders.
Minutes later, however, Anthony Gordon was hauled to the ground by Ashley Westwood and Richarlison sent England goalkeeper Nick Pope the wrong way from the penalty spot.
Then, after Alex Iwobi's pass split the Burnley defence, Vitaliy Mykolenko was tripped in the box and referee Mike Dean, after prompting from the VAR, awarded a second penalty spot that Richarlison duly converted.
Burnley equalised 11 minutes after the restart when Rodriguez turned in a cross that came through a crowd of players.
Pope then blocked Richarlison's overhead kick before Cornet became an increasing threat as rain swept over the ground.
The Ivory Coast international sealed victory in the 85th minute when another defensive blunder saw Ben Godfrey fail to deal with a cross and that allowed substitute Matej Vydra to drill a cross into the path of Cornet, who made no mistake.