After both sides suffered chastening defeats at the weekend, it was perhaps unsurprising that the first half started in subdued fashion.
It took until the 23rd minute for either team to have a shot on target, when a free-kick from deep fell invitingly for Antoine Semenyo and Bournemouth’s top goalscorer fired powerfully towards goal, but Jordan Pickford saved well.
Everton’s biggest first-half threat came from a pair of James Garner inswinging corners shortly before half-time. The first flicked off Alex Scott’s head and onto the top of the crossbar, before Thierno Barry almost nodded in the resulting set-piece from close range, but a combination of Djordje Petrovic and the post kept the ball out.
Junior Kroupi thought he had put Bournemouth ahead with the last kick of the half when he nipped in at the near post to beat Pickford. The offside flag went straight up, though, and replays showed the striker had strayed offside to meet Amine Adli’s flick-on.
It was only the second goalless first half in Everton’s past seven competitive away games, but the visitors almost went ahead 11 minutes after the break. Kiernan Dewsbury-Hall led a lightning-quick counter-attack and the ball broke to Barry in the penalty area, but Petrovic sprang off his line to deflect it behind.
The striker had another chance to score his first Everton goal after the hour mark, but he blazed over from Garner’s low cross.
The game opened up in the final 20 minutes as both sides looked for a winner, and it was Everton who finally broke the deadlock as Grealish scored his second Toffees goal. The Manchester City loanee took possession 25 yards out and drove towards the penalty area, with his right-footed effort taking a big deflection off Bafode Diakite to leave Petrovic stranded.
Iliman Ndiaye then could have sealed victory, but Petrovic beat away his fierce effort.
Bournemouth pushed for an equaliser but Everton held on, as Toffees boss David Moyes claimed his eighth victory over the Cherries - with seven of those to nil.
The visitors leapfrog their opponents in the PL to end the night in the top half, while Bournemouth slip to 14th.
Flashscore Man of the Match: Kiernan Dewsbury-Hall (Everton)
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