Thierno Barry tapped home after Tyrique George's shot squirmed beneath Lukáš Horníček early in the second half.
Harvey Barnes started strongly, whipping in a couple of dangerous deliveries - one which drifted inches wide of Jordan Pickford's far post.
Then a William Osula cross deflected off James Tarkowski and almost caught Pickford out, but the visiting goalkeeper adjusted his body quickly to save.
In the 16th minute, Joelinton fired in a shot on the turn but Tarkowski blocked bravely. From the resulting corner, swung in from the right by Lewis Hall, Dan Burn headed off target.
A Merlin Röhl effort flew over the bar for Everton, then Barry failed to connect with the ball when well placed.
Pickford had to push over another bending Barnes ball after it looped up off Jarrad Branthwaite, then Joelinton sent a piledriver from 20 yards narrowly wide.
On the stroke of half time, Harrison Armstrong squandered a wonderful opportunity for the Toffees as he miscued from close range.
But Barry could not miss when George's shot wasn't dealt with by Horníček, allowing the Toffees' French frontman to give his side the lead.
**Newcastle United:** Lukáš Horníček, Lewis Hall, Fabian Schär, Nico González, Joelinton, William Osula, Harvey Barnes, Anthony Elanga, Joe Willock, Dan Burn (c), Amar Dedić
**Subs not used:** Nick Pope, Sven Botman, Aladji Bamba, Yoane Wissa, Malick Thiaw, Sean Steur, Bazoumana Touré, Jacob Murphy, Jacob Ramsey