The hosts made three changes to the team who lost 1-0 at Liverpool on Sunday, with Kurt Zouma, Cenk Tosun and Ademola Lookman handed starts.
Rafa Benitez also made three changes to his Magpies line-up, with Jamaal Lascelles, Christian Atsu and Jacob Murphy recalled.
The Toffees made the brighter start of the two teams, but it was the visitors who took the lead with 19 minutes played, as Murphy marked his return with a deep cross from the left which was expertly finished by the arriving Salomon Rondon.
The home side responded to falling behind by forcing a number of corners, with Federico Fernandez making a crucial block to deny Gylfi Sigurdsson.
However, Everton drew level seven minutes before the interval as a corner was allowed to drift through to the back post where Richarlison made no mistake from close range.
Tosun then went close twice as the hosts finished the first 45 minutes strongly, with Martin Dubravka making a superb save to keep the Magpies level.
Just before the hour Atsu saw his effort saved by home keeper Jordan Pickford, while at the other end Richarlison shot wide when well placed.
The hosts brought on both Bernard and Theo Walcott as they tried to push for a winner, with the Magpies introducing both Ayoze Perez and Joselu.
Late on both Atsu and DeAndre Yedlin went close to grabbing a winner for the visitors, who in the end left Goodison Park with a valuable share of the spoils.