Roberto Martinez s team had slumped to within three points of the Premier League s bottom three after their recent struggles.
But goals from James McCarthy, Romelu Lukaku and Ross Barkley gave the Merseyside club a first home league win of 2015 to lift them six points clear of the drop zone.
A dismal day for Newcastle was summed up when captain Fabricio Coloccini was sent off with his side already two goals down in the second half.
Despite good results in the Europa League, including a 2-1 win against Dynamo Kiev on Thursday, confidence has been fragile at Goodison, where a demanding crowd have been quick to get on the team s back in recent weeks.
But Belgium striker Lukaku showed against Kiev how his power on the ball can pay dividends and his ability to hold up play was crucial in the opening goal in the 20th minute.
Phil Jagielka drilled a 50-yard pass straight through a vacant Newcastle midfield all the way to Lukaku, playing right up against Coloccini, and he held it up just long enough for McCarthy to charge on to.
The Republic of Ireland midfielder still had some work to do but was assisted greatly by Tim Krul who, despite having full view of the shot, embarrassingly dived the wrong way and watched as the ball went in past his outstretched feet.
It was the injury-troubled midfielder s first Everton goal at Goodison and only the second of his Toffees career.
Everton s on-loan Tottenham winger Aaron Lennon started to find more space to run into after the break but he was actually heading away from goal when Yoan Gouffran fouled him inside the area to concede a 56th minute penalty.
Lukaku stepped up to dispatch the spot-kick for his fifth goal in six league appearances against Newcastle.
If that needless foul was not bad enough Coloccini s lunge on Lennon, which earned a straight red card, compounded their miserable afternoon.
Barkley put the seal on a morale-boosting win for Everton in stoppage-time when he rounded Krul to score with the last kick of the game.