James Tarkowski's stoppage-time goal denied Bournemouth victory on Marco Rose’s home Premier League debut.
Alex Scott's sublime finish just before half-time put Bournemouth in front and they looked on course to take all three points before Tarkowski fired home an Everton equaliser.
Bournemouth were unchanged from the side that lost to Manchester City on the opening weekend, as James Garner replaced Hayden Hackney to partner Nottingham Forest transfer target Harrison Armstrong in Everton's midfield.
The visitors would have been in front early on but for some heroics from Bournemouth goalkeeper Dorde Petrovic who first saved Jarrad Branthwaite's header from a corner before making another fine stop to deny Iliman Ndiaye.
Pickford did well to deny Rayan at the other end before Petrovic made another stunning double save, first stopping a goal-bound effort from Tyrique George before getting up to make a point-blank save to deny Barry, who should have scored from less than six yards out.
Bournemouth punished the visitors shortly before half-time after some excellent work from Evanilson. The striker turned Tarkowski before cutting the ball across to Scott, who showed remarkable composure to take an extra touch and slot into the far corner.
Barry then missed the target from close range before he was denied again in the second half when Petrovic got down well to save after the striker - who was later replaced by Beto - had been played in by Armstrong.
Bournemouth nearly snatched a second late on, but Evanilson's effort deflected behind.
And they were left to rue that miss when Tarkowski was first to react after Bournemouth failed to clear a corner to smash home a late equaliser.
10: SAVE! Dewsbury-Hall delivers a fine inswinging corner which Branthwaite meets at the near post. His header is going in but Petrovic sticks out an arm and flicks it behind for another corner.
15: SAVE! Brilliant from Rayan down the right to beat Garner and hold him off before driving into the box. The Brazilian manages to get a shot away from a tight angle but Pickford is there to save with his legs.
20: DOUBLE SAVE! First Petrovic stops George's effort from the edge of the area, but only by pushing the ball into the path of Barry. Barry should score from a few yards out, but the Bournemouth goalkeeper is there again!
41: GOAL! Rayan picks up the ball on the edge of his own box and then plays a long pass forward to Evanilson, who turns Tarkowski inside out before finding Scott on the edge of the area. The Bournemouth midfielder then shows real composure, taking one touch to set himself before placing the ball with his right foot into the far corner.
50: SAVE! It's Barry once more who thinks he's scored, but Petrovic is there to deny him again.
79: SAVE! Pickford gets down quickly to make a smart stop. The ball lands at the feet of Evanilson, but the Bournemouth striker's close-range effort is somehow deflected behind for a corner.
90+1: GOAL! Hackney's corner is a poor one. It's straight towards Tavernier, but the Bournemouth winger doesn't communicate with Christie and they both challenge for the ball. Neither player clears it properly and it falls to Tarkowski, who powers his effort low along the ground and into the far corner.
Bournemouth boss Marco Rose:
"It was mostly about the result but as a coach you evaluate the performance as well and in most phases of the game it was really good.
"It was a quite open game, quite up and down. We can get better in some parts of the game. Today for me it was not even a set-piece for the goal because we had already cleared the situation, but I don't want to blame anyone because I saw so many positive things.
"I am really thankful for the warm welcome and I really appreciate to be coach of this team. It was a really nice feeling to coach this game.
"Like last week we would like to present our fans with a win but we will have to wait one more week."
Everton boss David Moyes:
"I was disappointed with the second-half performance. I was pleased with the first half and I don't think we deserved to be 1-0 down at half-time.
"I thought we'd done well enough but we missed big opportunities to make a better chance or score a chance when we had it in the first half, and I was hugely frustrated. I think the players were frustrated by it as well."
