The striker needed just two minutes to steer home Nikica Jelavic's cross to complete a comeback which saw the Toffees condemn Newcastle to a ninth defeat in 11 Barclays Premier League games.
Papiss Cisse headed the hosts in front with just 72 seconds played, but Leighton Baines' stunning free-kick cancelled out the advantage two minutes before the break and Anichebe grabbed the winner.
Cisse capitalised on poor defending to head over a stranded Tim Howard after Tim Krul's long clearance caught the visitors napping.
Marouane Fellaini forced a fine 24th-minute block from Davide Santon, while Steven Pienaar looked odds on to equalise eight minutes later, only for Krul to come out on top.
In the meantime, Alan Pardew's men went close to extending their lead when James Perch headed Vurnon Anita's free-kick against the post.
After Fabricio Coloccini was harshly adjudged to have fouled Fellaini, Baines took full advantage. The England left-back stepped up to blast the resulting free-kick past Krul from the best part of 30 yards, prompting celebrations among the travelling fans and a tirade of abuse aimed at referee Martin Atkinson from the locals.
Newcastle came agonisingly close to taking the lead when Shola Ameobi expertly controlled Sylvain Marveaux's 58th-minute ball over the top and stabbed it past the advancing Howard, but also just past the post.
And the home side were caught out two minutes later when Jelavic cut inside from the left and crossed for Anichebe to slot home at the near post.
Gabriel Obertan tested Howard at his near post with a 65th-minute shot after being set away by Shola Ameobi. But Krul had to tip Fellaini's audacious 71st-minute flick over the bar as Everton saw out the game.