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Before kickoff, both sets of players and supporters at Goodison paid tribute to the 96 Liverpool fans who lost their lives at Hillsborough Stadium in 1989.
It was recently found by a jury that the fans were unlawfully killed, and Everton decided to pay their respects to the victims and families, who were in attendance for the game.
After play got underway, the home side had the perfect start when Cleverley put them a goal to the good in the seventh minute.
The midfielder received the ball off a square pass from Ross Barkley before driving past the defender and drilling a left-footed shot into the far post.
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Everton complained to referee Neil Swarbrick that the goal should not have stood because Callum Wilson pushed debutant Matthew Pennington in the build-up, but the match official waved away their claims.
After equalising, Eddie Howe's Bournemouth team looked like the better side as their passing was slicker, while Everton struggled to cause trouble in their opponents' half.
The defender had to stretch to connect with the ball, and his header ended up drifting wide. Half an hour after scoring, Everton created just their second real chance of the game as Barkley forced Artur Boruc into a save.
At the stroke of half time, Martinez, whose future has reportedly been discussed by the Everton board, received a message from the supporters when a plane carrying a banner, which read 'TIME TO GO ROBERTO', flew over Goodison.
After the break, Everton improved on the ball and it took a brave block from Steve Cook to deny the home side a decent scoring opportunity.
Baines was part of the move as he sent a low cross into the area, looking for Barkley, who was closing in on the ball until Cook stepped in at the near post, and hurt himself in the process, but was given the green light by the medical staff to continue.
Bournemouth managed to put the hosts under pressure and Matt Ritchie had a decent chance from a free kick in Everton's half, and despite getting the ball over the wall, he failed to trouble Howard as he comfortably got hands to it.
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A matter of minutes later, Baines was celebrating Everton's second goal of the game when he received the ball after Aaron Lennon cut it across from the right flank. The left-back then drilled a well-hit shot past Boruc, who was nowhere near it.
Howe made a double change by bringing on Junior Stanislas and Benik Afobe, and both got involved quickly - the former let the ball run across him before letting one fly, but Everton blocked his attempt.
Stanislas tried his luck with a follow-up shot, but Everton cleared, but only to Ritchie, who went for goal from the edge of the box but once again, the hosts denied them a way through.
The Cherries kept up the pressure and Afobe should have drawn them level at 2-2 in the 78th minute when Wilson delivered a perfectly-weighted ball into the area to find the striker, but his header flew narrowly over the bar.
Bournemouth committed men forward in the closing stages of the match, but Everton were able to keep out their opponents and record a much-needed win.