The visitors needed three points to have a chance of survival and put on an excellent display.
Josh King gave the visitors the lead from the penalty spot after 13 minutes, but Everton drew level before halftime through Moise Kean as the home team enjoyed the majority of possession.
But after Dominic Solanke restored their lead, Stanislas gave Bournemouth a two-goal cushion late in the second half.
It was just a second win in 13 matches for Bournemouth, and a first away since victory at Chelsea in December. However, their nine losses on the road since then went a long way to sealing their return to the Championship.
Postmatch, Bournemouth manager Eddie Howe was unsure about his future with the club.
"It hurts. I'm trying to keep it together, I'm really sad for the fans of this club. They've not been here for the last part of the journey and that makes it harder," Howe told BBC Sport.
"It's a time for reflection, a time to see what the next step is. Everything at the club is based around staying in the Premier League and I'll speak to the people that I need to speak to.
"It's a very quick turnaround and we have to make the right decisions now."
He added: "It's small margins, there has been loads of moments during the season, it's never down to one thing. As the manager I make hundreds of decisions and I haven't been good enough this season.
"Throughout the season there are ebbs and flows, I'm not going to sit here and criticise the players in any way, I have to back them and support them. First I have to look at myself and that I will do.
"I always wanted to bring success to the club and to play an entertaining sort of football, that's always the aim."
Everton end their disappointing campaign in 12th position, with much work for manager Carlo Ancelotti to do in the short preseason before the start of the new campaign on Sept. 12.
The win was a first at Goodison Park for Bournemouth and a first home defeat for Everton since November, but that will be scant consolation for Cherries manager Eddie Howe.
Bournemouth were back in front before the break, as Diego Rico's free kick was brilliantly headed home by Solanke.
Everton did most of the pressing in the second half, but the visitors grabbed a third when Stanislas showed great determination to create a shooting chance and beat the poor dive of Pickford.