The Cherries opened the scoring after just five minutes when summer signing David Brooks opened his Premier League account, but Patrick van Aanholt's equaliser early in the second half looked to be enough to earn Palace a point.
However, the hosts were awarded a penalty with just three minutes remaining of normal time when Mamadou Sakho was adjudged to have caught Jefferson Lerma, and Junior Stanislas kept his nerve to secure all three points from 12 yards.
The victory means that Bournemouth have now picked up 10 points from a possible 12 in home games this season, whereas Palace remaining 13th following their fourth defeat of the campaign.
It was Bournemouth who made the brighter start to the match, and they were rewarded for their high-tempo opening after only five minutes when Brooks became the club's youngest ever Premier League goalscorer.
Palace quickly went in search of a response and came close through Luka Milivojevic's free kick before calling Asmir Begovic into action for the first time with Max Meyer's low drive from the edge of the area.
The home side enjoyed the better of the first half, though, and Wilson slammed one cross into the side-netting before Steve Cook nodded straight at the keeper shortly after the half-hour mark.
The Eagles were offering enough to suggest that they could get themselves back into the match, though, and Jordan Ayew offered a glimpse of that late in the first half when he fired over the crossbar.
Ayew turned provider early in the opening stages of the second half to set up James McArthur, who also failed to keep his shot down, but Palace had their equaliser just 10 minutes after the restart.
Van Aanholt collected Wilfried Zaha's pass inside the penalty area before cutting onto his weaker right foot and blasting a powerful effort into the roof of the net, beating Begovic at his near post.
Bournemouth squandered a chance to regain their lead as Wilson saw Joshua King's pass squirm under his foot when well placed inside the area, and that almost proved costly as Andros Townsend curled an effort narrowly wide from the edge of the area just three minutes later.
The late pressure came from the home side, though, and Simon Francis fired one long-range effort wide before Hennessey came to Palace's rescue twice in quick succession.
The first saw a mistake from Van Aanholt gift possession to Ryan Fraser, who raced through on goal before finding Wilson. The striker only had the keeper to beat, but Hennessey spread himself well and kept his side level with a big one-on-one save.
Moments later Wilson was the provider when his cross found Dan Gosling, whose firm header was destined for the corner only for Hennessey to again push it to safety.
There was nothing the Palace keeper could do about the winner, though, with referee Mike Dean awarding a penalty against Sakho having adjudged his marking on Lerma to be too robust from a free kick as the Palace defending caught the Bournemouth midfielder with a flailing arm.
With King off the field and Wilson having missed a penalty earlier in the season, substitute Stansilas took on the spot-kick duties and blasted his effort straight down the middle to secure a return to winning ways in the league after last weekend's 4-0 drubbing at the hands of Burnley.
There was a late flash point as Lerma, Cheikhou Kouyate and James Tomkins all picked up bookings following a brief melee, but there were no such fireworks at either end of the pitch as Palace fell narrowly short.
The Eagles next host newly-promoted Wolverhampton Wanderers on Saturday, whereas Bournemouth take on Watford at Vicarage Road having moved level on points with the Hornets this evening.
BOURNEMOUTH (4-4-2): Begovic; Francis, S Cook, Ake, Smith; Brooks (Gosling 58'), Lerma, L Cook, Fraser; King (Stanislas 84'), Wilson
CRYSTAL PALACE (4-3-3): Hennessey; Wan-Bissaka, Tomkins, Sakho, Van Aanholt; Meyer (Kouyate 84'), Milivojevic, McArthur (Schlupp 77'); Townsend, Ayew (Sorloth 82'), Zaha