Papiss Cisse, who is again the number one attacker at Newcastle with Demba Ba's move to Chelsea set to be complete soon, gave the Magpies the lead in just the second minute of the contest when he coolly headed over Tim Howard after Tim Krul's long clearance reached him on the edge of the area.
Luckily for the Toffees, Baines produced a brilliant 30-yard dead ball two minutes before the break after referee Martin Atkinson had questionably awarded a free-kick to Marouane Fellaini.
Everton had improved their output after they fell behind and their perseverance paid off when Anichebe, on minutes earlier for Steven Naismith, slotted home Nikica Jelavic's cross from the left after he had cut back inside.
The victory allowed David Moyes' Everton side to go fifth on the table where they now sit just three points behind third-placed Tottenham as Alan Pardew's Toon remained 15th and are only two points above the drop zone following a third straight loss.
Sports Direct Arena, Newcastle-upon-Tyne - 49,391