A largely flat first half without too many chances for either side saw the Toffees lead at the break, after Iliman Ndiaye converted a penalty three minutes before the interval conceded by Rayan.
But the Cherries came out with more purpose in the second half, and the game flipped on its head in three minutes.
Rayan first made amends for giving away the penalty by heading home for his second goal in as many games, before Amine Adli powered home after James Hill headed a free kick back across goal.
Jake O’Brien was sent off for the hosts five minutes later, and while the ten men of Everton pushed hard for an equaliser, the Cherries held them off to claim victory and to extend their unbeaten run to six in the Premier League.
Andoni Iraola made two changes to his team which drew with Aston Villa as Alex Toth came in for his first start for the Cherries since joining in January, in place of Lewis Cook.
Junior Kroupi dropped onto the bench, with Ryan Christie starting for the first time since he returned from injury.
The first two sights of goal went the way of the visitors, as Evanilson’s effort was comfortably picked up by Jordan Pickford before Christie had a shot blocked from range.
Everton had the ball in the back of the net in the tenth minute as Ndiaye flicked home from an Idrissa Gueye ball in after Djordje Petrovic punched clear a corner, but the flag was up for offside.
There were few chances to speak of for either side up to the half-an-hour mark when the hosts saw two good chances go begging.
Thierno Barry missed the target as the ball fell to him after Petrovic couldn’t hold onto a James Garner free kick from 35 yards.
The Serbian stopper then denied Ndiaye as the ball flashed across the face of goal, but Petrovic got across to save the Senegalese forward’s effort.
But the Toffees took the lead three minutes before the break.
It was a soft goal for the Cherries to concede as Rayan sought to bring the ball clear from deep in the Bournemouth half, Jarrad Branthwaite stole in and the Brazilian brought the defender down in the box.
Referee Andrew Madley pointed to the spot and Ndiaye placed his spot kick right in the bottom corner, beyond Petrovic’s dive despite the keeper going the right way.
That saw the hosts take the lead into the interval, and they had a good chance to double their advantage in the opening minutes of the second half.
A Petrovic pass out for Adrien Truffert was read by Ndiaye and he intercepted, his cross was half cleared and Christie was dispossessed by Kiernan Dewsbury-Hall, but James Hill got back to head his shot over the bar.
Everton went close again as Jimenez denied Barry a tap in after the forward found space in the Cherries box.
Truffert was not far away with a shot from range that went the wrong side of the post, before Rayan beat Vitali Mykolenko on the outside and went down under the challenge, but no spot kick was given.
Iraola went to his bench shortly before the hour mark as Kroupi and Enes Unal replaced Evanilson and Toth.
Moments later, the Cherries were level – and it was Rayan making amends for giving away the penalty.
The young Brazilian headed home brilliantly at the far post from a pinpoint Truffert cross from the left for his second goal in as many games.
Three minutes later, Bournemouth took the lead.
Scott delivered a free kick into the box from deep, Hill headed back across goal and Adli was there to touch home and wheel away to celebrate in front of the travelling Cherries faithful.
Everton were reduced to ten men with 21 minutes left to play as O’Brien was given his marching orders for bringing Adli down after Kroupi slid the Moroccan in on the counterattack.
The Toffees went in search of an equaliser despite being down to ten, and Ndiaye headed high and wide from a Mykolenko cross, before Beto headed at Petrovic.
Adli and Christie were replaced by David Brooks and Cook as the game entered the final ten minutes.
Branthwaite headed over from a Garner free kick, before Senesi blocked a Gueye shot behind.
The Toffees continued to press for an equaliser and saw much of the ball for the remainder of the game, but they weren’t able to find the equaliser, and the Cherries collected all three points.
Everton: 1, J. Pickford; 15, J. O'Brien (sent off, 69’); 6, J. Tarkowski (c); 32, J. Branthwaite; 16, V. Mykolenko (Keane 90+3’); 37, J. Garner; 27, I. Gueye; 19, T. George (Armstrong 62’); 22, K. Dewsbury-Hall; 10, I. Ndiaye; 11, T. Barry (Beto 62’ (booked, 66’)).
AFC Bournemouth: 1, D. Petrovic; 20, A. Jimenez (booked, 90+6’); 23, J. Hill; 5, M. Senesi; 3, A. Truffert; 8, A. Scott; 27, A. Toth (Kroupi 58’); 10, R. Christie (c)(Cook 80’); 21, A. Adli (Brooks 80’(booked, 81’)); 37, Rayan (booked, 57’)(Smith 90+3’); 9, Evanilson (Unal 58’).