Riding high from two successive wins to open their campaign, it was little surprise to see a front-footed approach from Everton.
They’ll feel they should’ve taken the lead inside the opening 10 minutes too, as Kiernan Dewsbury-Hall’s inswinging corner was flicked goalwards by the head of Jarrad Branthwaite, but a stunning one-handed save from Dorde Petrovic kept him at bay.
Bournemouth weren’t particularly perturbed by the Toffees flying out of the blocks and had a good opportunity of their own minutes later when Rayan angled in off the right and hammered an effort at the legs of Jordan Pickford from the corner of the six-yard box.
Petrovic was Bournemouth’s hero in the early stages, but he was so nearly left red-faced when he spilt Tyrique George’s strike to the feet of the in-form Thierno Barry, but he redeemed himself by standing tall to prevent Barry adding to his four goals in his previous two games.
Keeping the Cherries in the game, Petrovic gave his side the platform to keep probing for an opener, one which duly arrived shortly before half-time when, following some great hold-up play from Evanilson, Alex Scott capitalised on his lay-off to fire a deflected effort into the bottom corner.
Barry squandered a huge chance to level the scores on the stroke of half-time when he rippled the side netting on the back of a swift counter-attack, and he spurned an even better one three minutes after the restart when he capitalised on some dithering in defence from Bournemouth but could only fire straight at the legs of Petrovic.
There was a growing expectancy that Everton’s big chances to get back into the game had been and gone, with Rose’s men managing to largely stifle David Moyes’ side as the game grew older.
James Hill came close to adding a second for the hosts, but he couldn’t manage to bundle the ball over the line from close range.
They were even closer to doubling their cushion when Rayan forced Pickford into a fine one-handed save before a last-ditch block on the follow-up thwarted Evanilson.
Those misses proved costly in stoppage time, as a defensive mix-up from an Everton corner fell perfectly for Tarkowski, who drilled home on the half-volley to maintain their unbeaten start to the season.
Disappointing for Rose, but Bournemouth at least stretched their unbeaten home run to 10 games.
Flashscore Man of the Match: Adrien Truffert (Bournemouth)