Moise Kean initially put Everton in front in the 30th minute with his first Premier League goal, though he required a little luck, as he capitalised on Newcastle's inability to clear the ball and then shot under Martin Dubravka, who probably should have done better.
The in-form Dominic Calvert-Lewin got his 10th top-flight goal of the season early in the second half, but that two-goal cushion was insufficient.
Florian Lejeune was Newcastle's hero, first scoring with an acrobatic attempt after a Jordan Pickford error, then restoring parity with a volley that just about found its way over the line amid a scramble in the penalty box, both goals coming in stoppage time.
After six matches – five of which ended in defeat – without a victory, Bournemouth finally got back to winning ways against Brighton and Hove Albion.
Harry Wilson got the ball rolling with a driven effort from the edge of the box after Dominic Solanke teed him up in the 36th minute.
Pascal Gross then put past his own goalkeeper just before the break, diverting in after a Bournemouth corner caused havoc in the visitors' box.
Callum Wilson then wrapped up a comfortable win 16 minutes from time, rounding Mat Ryan and slotting in unchallenged, with Aaron Mooy's late effort a mere consolation in a 3-1 defeat for the Seagulls.
It was not enough to move Bournemouth out of the bottom three, though – Aston Villa's last-gasp 2-1 win over fellow strugglers Watford decisive in that sense, as Tyrone Mings got the all-important clincher.
There was no such drama at Selhurst Park, however, as Southampton managed to hang on to its 2-0 lead in an ill-tempered affair, moving it into the top half of the table.
Palace's Wilfried Zaha might find himself in danger of retrospective action as well, after television cameras caught him poking James Ward-Prowse in the face during a half-time altercation.