James Tarkowski's foul on Sandro Tonali handed England winger Gordon the chance to score against his former club on Merseyside, but his international teammate was equal to it, saving low to his left.
That meant that Eddie Howe's side had to settle for a point in the end on their final trip to Goodison Park, after a hard-fought encounter with Sean Dyche's men.
In a largely even first half, United looked likelier to break the deadlock. Ten minutes into the contest, Bruno Guimarães met a Sandro Tonali flick-on at the back post with a controlled first-time volley, but saw it cleared the off the line by Iliman Ndiaye.
Dominic Calvert-Lewin saw an effort saved comfortable by Nick Pope at the other end and, after Jacob Murphy had tested Pickford, the Toffees had the ball in the net through Abdoulaye Doucouré. The midfielder's header from James Garner's fine cross from the right flew into the top left-hand corner, but he was well offside, and a relatively swift VAR check confirmed it.
The main flashpoint came on the half hour, when centre back Tarkowski inexplicably wrestled Tonali to the floor off the ball as the hosts were preparing to defend a cross from the right. Referee Craig Pawson awarded the spot kick after consulting his pitchside monitor and Gordon stepped up, but his low strike was kept out by Pickford, who dived to his left to repel it with his legs.
United responded well in the early exchanges after the break, with Tonali seeing a shot blocked before they squandered a big opportunity ten minutes after the restart, with Guimarães scooping a shot over the bar with his unfavoured left foot when well-placed.
Miguel Almirón was sent on for Harvey Barnes just after the hour, and the Paraguayan's deflected cross almost wrong-footed Pickford immediately after his arrival. But Everton felt aggrieved not to have a penalty of their own moments later, when Calvert-Lewin forced a smart save from Pope and attempted to convert the rebound, only to catch Dan Burn in the process. Pawson, though, was unmoved.
Almirón then saw a volley deflected wide by Tarkowski as the visitors continued to dominate in the closing stages, and inside the final ten minutes Newcastle had an even bigger chance to score. It fell to Gordon, who was played in by Almirón's neat through ball, but he couldn't keep his shot down with Pickford's goal beckoning, and the Toffees survived.
