Cleverley opened the scoring inside ten minutes but the Cherries hit back almost immediately through Marc Pugh, before Leighton Baines sealed the win with his first goal in over a year.
Explosive two minutes in a closely fought half
The home started slowly in the opening minutes of the game, however Everton managed to open the scoring after just seven minutes.
From a throw they built up play well and Cleverley picked up the ball from Ross Barkley on the edge of the area, who turned well before beating Simon Francis by stroking his shot across goal and into the bottom right corner.
It was the goal Everton needed yet the lead didn't last long as two minutes later Bournemouth got themselves level. A long ball down the line saw Callum Wilson get in behind the Everton defence after a slight push on young defender Matthew Pennington, his cut back saw Joshua King direct the ball to Pugh and he calmly fired past Tim Howard.
Both sides continued to cancel each other out which meant chances in the first half were hard to come by. Matt Ritchie threatened for Bournemouth with two efforts, unfortunately for the away side his first was brilliantly blocked on the edge of the area and his second attempt flew well over after taking a deflection.
For the hosts Barkley had two chances inside a minute, the first saw him burst forward and after being directed across the top of the area he could only fire over the bar.
Shortly after, Oumar Niasse slipped the ball through to Barkley and his attempted poke past Artur Boruc was well saved by the Bournemouth keeper as he was quick off his line to deny the English midfielder.
Baines blasts Everton into the lead
The second half started the same as the first ended, with both teams trying to create chances but failing to do so. The first noteworthy attempt of the half saw King in plenty of space and he unleashed a low effort at goal, which fortunately for Howard rolled wide of the post.
Bournemouth had a chance to threaten with a free-kick, however it was a tame effort and Howard comfortably caught the ball.
Moments later Everton regained the lead, Barkley showing brilliant strength to hold off his marker before charging forward and playing the ball wide to Aaron Lennon. The winger's cross found its way to Baines on the other side of the area and the left-back fired a great strike into the roof of the net.
Bournemouth reacted well to going behind and they built up pressure on Everton with a number of crosses and great build-up play around the area, but they weren't able to create anything as the Everton defenders remained strong and repeatedly threw their bodies on the line.
Toffees hold on despite Afobe scare
The away side soon managed to create a golden chance. Wilson beat his man and charged to the byline and his drilled cross found the head of Benik Afobe, whose header flew just over Howard's bar.
A few minutes from the end Bournemouth's Ritchie fired from range once more and this time it found its way past the Everton defenders and it flew just wide. Despite their huffing and puffing Bournemouth couldn't find the equaliser as Everton defended well to see out the three points.