Thierno Barry scored the decisive goal shortly after half-time as David Moyes' side continued their impressive recent form and moved up to seventh in the Premier League table.
The result came just four days after Everton defeated Newcastle 3-1 in the Carabao Cup, giving the Toffees another victory over Matthias Jaissle's team.
Everton started brightly and created the better opportunities in the opening half. Barry missed a promising chance from close range, while Merlin Röhl fired over the crossbar.
Newcastle struggled to impose themselves on the contest, although Joelinton came close with a powerful effort that narrowly missed the target. Harrison Armstrong also missed an opportunity for Everton in first-half stoppage time.
Neither side managed a shot on target before the break, with Newcastle's supporters expressing their frustration as the teams headed into the dressing rooms.
The breakthrough arrived in the 49th minute when Tyrique George's effort slipped through goalkeeper Lukas Hornicek's hands. Barry reacted quickest to turn the loose ball into an empty net and give Everton the lead.
The goal was Barry's eighth across all competitions this season and his third consecutive match scoring against Newcastle.
Everton pushed for a second, with Hornicek making important saves to deny Jarrad Branthwaite, Kiernan Dewsbury-Hall and Jake O'Brien.
Newcastle improved as the match progressed and nearly equalised when Anthony Elanga's effort struck the post before rolling across the goal line and being cleared by the Everton defence.
Jordan Pickford also produced an important late save to deny substitute Yoane Wissa, helping the visitors protect their advantage.
Newcastle's difficult evening ended on another negative note when substitute Jacob Ramsey received two yellow cards in quick succession and was sent off in the eighth minute of stoppage time.
The defeat was Newcastle's fourth in their last five matches across all competitions, increasing the pressure on manager Jaissle.
Everton, meanwhile, climbed to seventh place after ending their nine-game winless run away from home in the Premier League.
The victory also marked the first time Everton had won three consecutive Premier League away matches against Newcastle.
Final score: Newcastle United 0-1 Everton
Goalscorer:
• Everton: Thierno Barry 49'
Red card:
• Newcastle United: Jacob Ramsey 90+8'
Man of the Match: Kiernan Dewsbury-Hall (Everton)