Iliman Ndiaye handed the hosts the lead at the end of the first half after Rayan had given away a penalty, but the Brazilian responded with the first of the Cherries goals shortly after the hour mark.
Match report and reaction:
Amine Adli then netted three minutes later to hand Bournemouth a lead they would not relinquish and they returned to Dorset with the points.
The Daily Echo rates the Cherries in victory.
Djordje Petrovic: 7
A solid performance from the Serbian, who has certainly improved of late. His command of the box was good, coming to punch on occasion, and he made some good saves, most notably from Ndiaye in the first half.
Bournemouth goalkeeper Djordje Petrovic clears the ball during the Premier League match at the Hill Dickinson Stadium, Liverpool. (Image: Peter Byrne/PA Wire)
Adrien Truffert: 8
A beauty of an assist for Rayan, a wonderful cross with his weaker right foot. Otherwise good defensively, and he continues to be a threat going forward.
Marcos Senesi: 6
A decent performance from the Argentine.
James Hill: 9
My man of the match. Hill continues to impress at centre back and he was colossal again at Everton – and he picked up another assist for Adli’s goal.
Alex Jimenez: 7
The Spaniard, who is set to join the Cherries on a permanent move after the obligation in his loan deal was triggered, was a livewire. Always busy going forward, he also had some key interventions at the back as he denied Barry a certain tap in in the second half.
Alex Scott: 7
Scott continues to impress in midfield and it was another top performance. He’ll be hoping Thomas Tuchel has been keeping a keen eye on Cherries games of late!
Bournemouth's Alex Toth (left) and Everton's Vitaliy Mykolenko (right) battle for the ballduring the Premier League match at the Hill Dickinson Stadium, Liverpool. (Image: Peter Byrne/PA Wire)
Alex Toth: 6
A difficult first start for the Hungarian. While he didn’t do anything wrong, he wasn’t able to stand out in what was a flat game before he was replaced shortly before the hour mark.
Amine Adli: 8
A well-taken goal and good work to draw the foul that saw Jake O’Brien sent off. Adli looks more and more comfortable each week he plays in the side.
Ryan Christie: 7
A good showing from the Scot in his first start since returning from injury. He looks to be returning to somewhere close to his levels from last season, which is a huge positive for the Cherries.
Bournemouth's Rayan celebrates scoring their side's first goal of the game during the Premier League match at the Hill Dickinson Stadium, Liverpool. (Image: Peter Byrne/PA Wire)
Rayan: 8
The Brazilian must have been gutted to give away the penalty in the first half with perhaps some naivety in only his second start and third appearance in the Premier League. But where other 19-year-olds may have gone back into his shell, he improved in the second half and scored a marvellous header, becoming only the third player in Premier League history to have three goal involvements in his first three games, after Robbie Keane and Anthony Martial.
Evanilson: 5
Like Toth, a tough game for the Brazilian before he was replaced. A lot of running as usual but in a flat first half he struggled to make an impact before he was replaced.
Substitutes:
Enes Unal: 5
Junior Kroupi: 7
Lewis Cook: 6
David Brooks: 6
Adam Smith: N/A