James McCarthy gave the Toffees the lead in the 20th minute on Sunday and Romelu Lukaku doubled the advantage with a penalty shortly after half-time.
Newcastle had barely recovered from going 2-0 behind when captain Fabricio Coloccini was shown a straight red card for a reckless tackle on Aaron Lennon.
And Ross Barkley rounded off a fantastic afternoon for Everton when he netted the third deep into injury-time.
The win is a welcome relief for Everton manager Roberto Martinez after a disappointing run of results at home in the league.
The Toffees’ previous victory Goodison in the league came when they saw off QPR 3-1 on 15 December, and they did show signs of nerves against Newcastle.
The visitors started brightly and almost took an early lead as Gabriel Obertan’s close-range shot was kept out by Tim Howard and Mike Williamson’s follow-up was blocked by McCarthy.
But Everton soon started to assert themselves and, after Lukaku tested Tim Krul with a couple of long-range efforts, it was McCarthy who broke the deadlock on 20 minutes.
The midfielder still had some work to do after collecting Lukaku’s lay-off but was assisted greatly by Krul who, despite having full view of the shot, dived the wrong way and watched as the ball went in past his outstretched feet.
But Everton were unable to build on the goal and instead it was Newcastle who finished the half stronger.
Sammy Ameobi had a couple of efforts blocked while Emmanuel Riviere should have done better after beating Howard to Moussa Sissoko’s cross.
Newcastle continued to look the better side after the break but the momentum swung firmly in Everton’s favour as referee Martin Atkinson awarded a penalty and showed Coloccini a red card inside a decisive five-minute spell.
Yoan Gouffran gave away the spot-kick after failing to clear the ball inside the area and then bringing down Lennon after he was tackled.
Lukaku converted the penalty and it got worse for Newcastle minutes later as Coloccini dived in recklessly on Lennon and was shown a straight red card.
Everton spurned chances to add to their advantage as Arouna Kone and Leon Osman were denied by Krul and Barkley had a shot blocked on the line by Ryan Taylor.
And Newcastle came close to setting up a nervy finish as Howard produced a brilliant stop to keep out Sissoko’s shot and Antolin Alcaraz blocked Remy Cabella’s driving run.
But Everton made it 3-0 deep into injury-time as Christian Atsu played a ball through for Barkley and he rounded Krul and finished into the empty net.
Everton: Howard (6), Coleman (6), Alcaraz (7), Jagielka (7), Baines (7), McCarthy (7), Gibson (6), Osman (6), Lennon (7), Kone (6), Lukaku (7).
Subs: Atsu (6), Barkley (7), Besic (5).
Newcastle: Krul (5), Taylor (5), Williamson (5), Coloccini (4), Janmaat (5), Colback (6), Gouffran (5), Obertan (5), Ameobi (5), Sissoko (6), Riviere (5).
Subs: Cabella (5), Gutierrez (5), Perez (6).
Man of the match: Aaron Lennon.