Abdoulaye Doucoure thought that he had sent Everton ahead in the 18th minute of the contest through a header, but his effort was disallowed for offside.
Newcastle then had the chance to score from the penalty spot in the 35th minute through Gordon after James Tarkowski had hauled Sandro Tonali to the ground from a corner.
However, Gordon saw his effort kept out by Everton goalkeeper Pickford, which sparked wild celebrations inside the stadium, with the former passing up the chance to register against his former club.
Newcastle dominated the first period, boasting 71% of the ball, but their only attempt on target came from the penalty, with the Magpies struggling for inspiration in the final third of the field.
Everton wanted a penalty of their own in the 66th minute when Dominic Calvert-Lewin went down inside the box under a challenge from Dan Burn, but the referee was not interested and VAR agreed.
Newcastle were the team pressing for a winner in the final exchanges, but Eddie Howe's side were unable to make the breakthrough, with the points being shared at Goodison Park.
The result has moved Newcastle into sixth position in the Premier League table, boasting 12 points from their opening seven matches, while Everton remain 16th, collecting five points from their first seven games.
EVERTON VS. NEWCASTLE HIGHLIGHTS
Anthony Gordon penalty miss vs. Everton
Gordon has the chance to send Newcastle ahead from the penalty spot against his former club, but Everton goalkeeper Pickford is on hand to make a smart save.
Tough moment for Gordon, but the Everton supporters certainly enjoyed that one.
MAN OF THE MATCH - JORDAN PICKFORD
Pickford was the Everton hero on Saturday, keeping out Gordon's first-half penalty, which was as good as a goal, while he made a number of other excellent saves at Goodison Park.
The 30-year-old made three outstanding stops during the goalless draw, and Everton certainly have their goalkeeper to thank for securing a share of the spoils against the Magpies.
EVERTON VS. NEWCASTLE MATCH STATS
Possession: Everton 32%-68% Newcastle
Shots: Everton 8-16 Newcastle
Shots on target: Everton 2-3 Newcastle
Corners: Everton 0-10 Newcastle
Fouls: Everton 12-8 Newcastle
WHAT NEXT?
Everton will be bidding to stretch their unbeaten run in the Premier League to four matches when they head to Ipswich Town in their first match after the international break.
Newcastle, meanwhile, will aim to return to winning ways when they welcome Brighton & Hove Albion to St James' Park immediately after the international break.