A beautiful ball from Lewis Cook played Stanislas in behind the Stoke defence, leaving him only to calmly slot it past Joe Bursik.
In what was a lively first half, Stoke were the better team, coming close on numerous occasions, most notably from a Jordan Thompson free-kick which hit the side netting.
Bournemouth did have the better of the chances though, also hitting the crossbar when Dominic Solanke lobbed the keeper, only for it to bounce off the woodwork and over.
This result means Stoke stay in seventh, while Bournemouth moved up two places to third, two points off the automatic promotion spots.
Story of the match
Stoke came into this game two points off the play-offs are the earlier results in the day, with only one loss in their last six in the league. Stoke made two changes from Tuesday's 1-1 draw with Nottingham Forest, with John Obi Mikel and Steven Fletcher starting in place of Jacob Brown and Sam Vokes.
Bournemouth also came into this game with one loss in their last six, finding themselves in fifth place after results earlier in the day. the Cherries also made two changes, with David Brooks and Philip Billing being omitted from the starting eleven, the latter was out due to a groin strain. Diego Rico and Sam Surridge took their places.
The first piece of action came from the away side, when Rico fizzed a low cross to Surridge, who hit it wide with his left foot, a chance he probably should have put on target.
At the 15 minute mark, some good pressing from Stoke allowed Fletcher to give the recently returned Joe Allen a shot, that ultimately was comfortable for Asmir Begovic.
However, the former Stoke keeper was almost caught off guard soon after, when Thompson whipped a free-kick into the side-netting, not too far from giving the Potters a relatively early lead.
Bournemouth then hit the woodwork, Solanke hitting the crossbar with a chip shot, after Lewis Cook played him in behind the Stoke defence with a beautiful pass on the turn. Arguably a shot Solanke should have put away, especially as Bursik stuttered as to whether to close him down or not.
25 minutes in and the home side were forced into their first change, Fletcher coming off with what seemed to be a muscular injury after landing awkwardly, and was replaced by Brown. A blow for the Potters, as he had played well up to this point, the only player to claim to have a 100% pass accuracy in the game.
Tommy Smith had been a nuisance for Bournemouth down their left-hand side, and the free-kick he won lead to a Nick Powell header that was glanced wide, another Stoke chance that went begging.
Bournemouth then had their chance from a free-kick, when Stanislas curled one around the wall and forced Bursik to push it wide.
The Cherries were probably the worse of the two sides in the first half but started the second well, Rico putting another cross in the box, and it went across the six-yard box where Bursik intervened with a push, stopping a Bournemouth player from getting on the end of it before the captain James Chester cleared.
In a quiet second half, Stoke did put the ball in the net, however, Powell appeared to shove Steve Cook in the build-up, meaning a free-kick was given the other way.
Bournemouth then took the lead through Stanislas, when Lewis Cook played a brilliant ball over the top again and in behind for Stanislas, who calmly slotted it past Bursik to give Bournemouth a goal they cherished.
Adam Smith then hobbled off injured with what looked to be a knee injury after a clash with Brown, an injury that gives Bournemouth fans cause to concern.
The cherries then nearly sealed the victory after the ball was played across the box and ended at the feet of Stanislas, only for his shot to be blocked.
With five minutes added on, Stoke pushed for an equaliser but was unable to find one, meaning they missed out on an opportunity to go into the play-off places, and level on points with Bournemouth.
Man of the match: Junior Stanislas (Bournemouth)
The only goalscorer, it had to be given to him. The man who provided a real difference, he moved to centre attacking midfield from left wing-back and suited the change in role perfectly, making brilliant runs and causing Stoke many problems, and a well-taken finish too.