The Toffees were at risk of a relegation battle after failing to win over of their previous five games, but the visitors, who had Fabricio Coloccini sent off, were easily beaten
Roberto Martinez's men came into Sunday's meeting with just one victory in 12 in the league but errors from Tim Krul and Yoan Gouffran handed them the first two goals, before Newcastle captain Fabricio Coloccini was sent off and substitute Ross Barkley added a late third.
Everton took the lead midway through the first half when Newcastle goalkeeper Krul - who gifted Manchester United's winner to Ashley Young last time out - let James McCarthy's weak shot from 20 yards into the net.
Then, with 10 minutes of the second half gone, Gouffran was dispossessed inside his own penalty area and his subsequent foul on Aaron Lennon allowed Romelu Lukaku - scorer of a late penalty in Thursday's UEFA Europa League win over Dynamo Kiev - to once again convert from 12 yards.
Shortly afterwards, Coloccini was perhaps harshly shown a straight red card by referee Martin Atkinson for a foul on Lennon and Barkley's cool finish in stoppage time completed a miserable day for Newcastle.
With victory, Everton lifted themselves six points clear of the drop zone, while Newcastle remain 11th, four points better off.
Newcastle came close to opening the scoring in the third minute as Everton failed to clear a corner.
The ball fell to Gabriel Obertan, whose shot was parried by Tim Howard, and Mike Williamson's rebound was kept out on the line by McCarthy, with the visitors appealing in vain for handball.
Lukaku then tested Krul with a low shot three minutes later as both sides started with purpose.
With 20 minutes gone, Everton took the lead as McCarthy picked up Lukaku's lay-off and hit a low shot that Krul somehow allowed to creep under his legs as he dived in the wrong direction.
Newcastle enjoyed a sustained period of pressure as the interval approached but, with the final action of the half, Everton almost doubled their advantage when Antolin Alcaraz's header from a corner was cleared from just in front of the goalline.
Ryan Taylor curled a free-kick narrowly over the crossbar early in the second half but in the 56th minute Lukaku put his side further ahead.
After being robbed by Lennon inside the box, Gouffran brought down the winger with a clumsy challenge and Lukaku sent Krul the wrong way from the spot for his first league goal since the end of January.
Things went from bad to worse for the visitors three minutes later as Coloccini was shown a straight red card for a lunge on Lennon.
Arouna Kone, Leon Osman and Barkley were all denied by Krul as Everton went in search of a third, before Howard produced a fine save to keep out Moussa Sissoko's low shot 12 minutes from time.
And in the final minute of stoppage time Everton broke upfield and Christian Atsu fed Barkley, who rounded Krul and slotted home to seal a comprehensive victory.