The Whites and the Toffees battled to a 2-2 draw in LS11 on Saturday afternoon with Rafa Benitez's visitors twice taking the lead in West Yorkshire.
Dominic Calvert-Lewin opened the scoring from the spot after VAR deemed Liam Cooper's challenge to be a penalty on pitchside review by match official Darren England.
Mateusz Klich levelled before the interval to set United alight but Gray put the Merseyside visitors ahead once more after the break.
England striker Calvert-Lewin then saw a glorious one-on-one chance go begging thanks to the onrushing Illan Meslier, an effort which would've handed Everton a two-goal cushion.
The visitors were made to pay in the 73rd minute as a low finish from winger Raphinha hauled the hosts level and saw Marcelo Bielsa's side earn their first point of the season in front of a full capacity crowd.
Gray - who caused problems all afternoon down both flanks - was left pleased with a battling showing from the Toffees but admitted he and his team-mates need to have a more killer instinct when opportunities arise.
“It’s important to me to make a good start at Everton," he said post-match.
"I feel I’ve integrated into the team and the system and I'm working well with the manager. My focus is to keep being a threat and hopefully put more goals in.
“Against a team like Leeds you have to kill the game off. They’re always going to apply the pressure and with the fans here, they’re not going to give up. We could have taken more today but we’ll take the point.”