The Magpies were awarded a spot kick for a second consecutive week when James Tarkowski dragged down Sando Tonali inside the box on the half hour mark.
Gordon stepped up to take it in Alexander Isak's absence but saw his effort saved by former Toffees team-mate Pickford, who dived to his left.
The England international goalkeeper said after the game that he knew where Gordon was going to place the penalty once it was awarded.
'I watched the game last week and and he reversed it against Ederson and the one before he went the same way so I fancied him going keeper's left, so I said that I'm going to go keeper's left which I did,' Pickford told Sky Sports.
Anthony Gordon's first half penalty was saved as Everton draw 0-0 with Newcastle
'[It] hit me off my knee. I don't think Ant executed it to perfection and he'll be disappointed but I'm there to make the save and that's what I did.'
Asked whether it helps playing against a former team-mate, he added: 'Not really, we try and double bluff each other and I've got the better of him tonight.
'Pleased because it's a big moment in the game and it managed to help us draw and we move forward.'
Pickford and Gordon were seen conversing shortly after the game ended and the former revealed what they talked about.
'He just asked me why I went that way and I just said, "You went right the last two times so I fancied you going left", which I did,' Pickford said.
'I back myself on pens and I think it's my first one for Everton in a while. It's pleasing, it's a clean sheet and we build on that.'
The draw at Goodison Park has ensured Everton stay clear of the relegation zone, with Sean Dyche's side two points ahead of 18th-placed Crystal Palace.
Meanwhile, Newcastle missed an opportunity to go into the top four following a second straight draw.