Stanislas, who had scored a 98th minute equaliser against Everton in a two-goal late show in last season's equivalent fixture, struck after 23 minutes with a magnificent 25-metre shot.
Bournemouth were well worth their lead after England international Jack Wilshere, on loan from Arsenal, had earlier had a shot deflected off Seamus Coleman onto the bar.
The home side dropped deeper after the break as Everton applied the pressure but Ronald Koeman's men were uninspired as the Cherries kept only their second clean sheet in their last 15 league games.
(Reporting by Ian Chadband; Editing by Rex Gowar)
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