The visitors took the lead through a Josh King penalty, after Richarlison handled the ball in the area, early in the first-half, but Everton drew level through Moise Kean in the 41st minute.
Dominic Solanke headed the visitor ahead before half-time and Junior Stanlislas doubled their advantage late in the game, but it wasn't enough as Aston Villa picked up a point against West Ham United.
Story of the Game
The visitors knew just three points alone wouldn’t be enough to keep them in the top-flight. The last team to enter the final day in the relegation zone and survive was Wigan Athletic back in 2011. Nevertheless, it was Bournemouth that showed their attacking intentions from the start.
With just six minutes on the clock, a cross was swung to the back post of the Everton box. The ball ricocheted and fell to the feet of King, but Jordan Pickford was equal to the shot at his near post. Down the other end went the Blues though and Coleman was then denied by Aaron Ramsdale.
King and Solanke linked up well to release Callum Wilson and the striker tried to loft the ball over the approaching Pickford, who stood strong and made a terrific save.
Three minutes later, the visitors were awarded a golden opportunity to go ahead as Wilson’s header bounced up and hit the arm of Richarlison in the area. Wilson himself stepped up and calmly stoked the spot-kick into the bottom corner, past the outstretched arm of Pickford.
There were cries for another penalty from the Bournemouth players. David Brooks’ mishit shot came off the arm of Lucas Digne, who was turning away from the shot, and VAR deemed the incident as a ball to hand on the French full-back.
Everton squandered a great chance to draw level on 19 minutes when Kean played a perfectly weighted through ball to the advancing Walcott. The former England international raced into the area and was denied at the near post by the legs of Ramsdale.
The Blues enjoyed a spell on top after the half-hour mark and applied pressure as they went in search of an equalising goal. Richarlison was unable to generate enough power on Walcott’s cross to trouble the ‘keeper, while Gylfi Sigurdsson fired over the bar from the edge of the area.
Four minutes before the interval, that equaliser did come and it was a blow for the visitors. Coleman’s reverse pass played Walcott through down the right and his low cross picked out Kean at the back post for his second goal of the season.
Kean threatened to turn the game right around inside the space of a few minutes as his header was saved by Ramsdale, but the Cherries ensured they’d be going into half-time ahead as they retook the lead from a set-piece in additional time.
Rico whipped a free-kick into the box from the left side and Lerma guided his header across goal and into the far corner of the net. With Watford trailing 3-1 against Arsenal at the break, Bournemouth knew they needed a West Ham goal against Villa in the second-half if they were to stand a chance of avoiding the drop.
Bournemouth picked up where they left off in the first-half and were denied a third goal on 48 minutes. Ramsdale cleared the ball forward and Wilson neatly brought the pass down before volleying at goal, which was pushed away from danger by Pickford.
Just after the hour-mark, Digne and Kean linked up as the full-back played a cross into the box, but this time the Italian forward was unable to generate power onto his header, which was gathered by Ramsdale.
Everton were probing away for an equaliser and were patient with their build-up, but the visitors continued to look threatening on the counter attack.
Leighton Baines, who announced his retirement after the game, made a goal-saving last-ditch challenge with 15 minutes to go to deny Wilson. However, the Cherries pursued and doubled their lead five minutes later.
The visitors failed to clean Rico’s set-piece and King combined with substitute Stanislas, who broke into the area after cutting inside and fired across into the far corner of the net.
With five minutes to go, the dejected look on the Bournemouth bench after Jack Grealish’s goal filtered through soon turned into a look of hope after news that Andriy Yarmolenko equalising just one minute later.
The goal Bournemouth needed didn't come and their fate was confirmed when the full-time whistle went at the London Stadium. The Cherries will join Norwich and Watford in the Championship next season.
Man of the Match: Josh King
King caused problems all afternoon for Everton and was the focal point for most things going forward for Bournemouth at Goodison. King got the scoring underway from the penalty spot early on and provided the assist for Stanislas' goal, which secured the victory.
What's next for the two sides?
It could well be a summer of change for Everton as Ancelotti looks ahead to his first full season in charge of the Blues. Departures and arrivals are a certainty and the Italian boss will want a much better season and aim to qualify for Europe.
Meanwhile, Bournemouth will prepare for life in the Championship and Howe may well see a few players depart the south coast to stay in the top-flight as the likes of Nathan Ake has already been linked with a move to Manchester City.