Burnley took the lead after 12 minutes as Cornet’s corner was volleyed in by Collins at the far post. But the hosts’ advantage lasted less than six minutes as Richarlison drew Everton level from the penalty spot after Anthony Gordon was fouled inside the area by Ashley Westwood.
A second penalty from the Brazilian put The Toffees 2-1 ahead at half-time following a VAR check for a foul by Aaron Lennon on Vitalii Mykolenko.
After the restart, Rodriguez drew Burnley level by turning in a low Charlie Taylor cross.
And with just five minutes of normal time remaining, Cornet secured a crucial three points for Burnley to make it 3-2.
The win sees The Clarets move above Watford and up to 18th on 24 points, just a point behind Everton in 17th.
The Toffees would have moved six points clear of the bottom three but instead they have Burnley breathing down their neck heaving into a tough run of fixtures.
Burnley’s win also means Newcastle sit seven points above the relegation zone with eight games left to play while both Burnley and Everton have a game in hand on them.
Next up for Newcastle is a home match against Wolverhampton Wanderers while Everton host Manchester United and Burnley are away to Norwich City.