Stanley made a rapid-fire start with three goals in the opening 20 minutes through Paul Mullin, Terry Gornell and Rostyn Griffiths.
This was only the Reds third home win of the season, and the Cherries are still deep in trouble having not yet reached positive points after a damaging pre-season deduction.
Veteran striker Mullin, the club's captain and record goal-scorer, set Stanley on their way with a neat strike into the bottom right corner in the third minute.
Strike partner Gornell got his fourth goal of the season when his long-distance drive flew high into the centre of the goal ten minutes later.
And just before the 20 minute mark it was game over when on-loan Blackburn midfielder Griffiths, an Australian, fired home a 25-yarder into the top right corner.
The visitors did manage to exert some pressure later on in the game, but it was too little too late.
This was a performance that will delight Stanley manager John Coleman, who has often been shell-shocked by one dismal home showing after another.
But there was never any danger of them losing this clash, despite the facts showing that Bournemouth actually had more shots on target.
Earlier in the season, Accrington lost a home game 3-2 to Bradford after being 2-0 up with ten minutes to go.
But, despite some visiting pressure, the Reds defence held firm right until the final whistle and a clean sheet will please Coleman just as much as the three goals.