The hosts at the Vitality Stadium opened up a two-goal advantage after 23 minutes as Koby Mottoh and Michael Dacosta Gonzalez found the back of the net.
But the Baggies were back on level terms before the interval as a Kai Crampton own goal and an Eseosa Sule strike saw the sides enter the break at 2-2.
West Brom edged in front when Sule applied the finishing touch from a corner that Bournemouth failed to deal with, before the visiting captain completed his hattrick with a finish across Crampton.
Remy Rees-Dottin narrowed the deficit for Bournemouth, before Noah Dupont restored the two-goal lead for the visitors.
Eugene Nyarko gave the Cherries hope in stoppage time but they weren’t able to find an equaliser to send the tie to extra time and they were beaten 5-4.
Koby Mottoh opened the scoring for the Cherries. (Image: Robin Jones/AFC Bournemouth)
First team player Ben Gannon-Doak started for Alan Connell’s side as he continued his return from injury, under the watchful eye of Andoni Iraola, the boss in attendance at Dean Court.
The Cherries took the lead with the first shot on goal in the ninth minute.
Tonks collected a pass from Crampton, and he played an excellent ball in behind for Nyarko to charge on to.
Nyarko played a pass across the box to Mottoh, and he showed good composure to finish into the bottom corner.
Dacosta Gonzalez tested Ben Cisse from the angle shortly after the goal, before he again had a sight of goal with a volley from the centre of the box, but, while it was powerfully hit, it was well blocked before it could trouble the keeper.
The Baggies’ first sight of goal came in the 17th minute when George Shaw shot over the bar after a corner came out to him.
Koby Mottoh (Image: Robin Jones/AFC Bournemouth)
Dacosta Gonzalez doubled the Cherries lead with 23 minutes on the clock as he picked up a loose ball and powered towards goal with his left from 25 yards, and he found the bottom corner via a deflection.
But the deficit was halved three minutes later.
A free kick from Alfie Maughan was met by Shaw and his header hit the upright and bounced back across goal and Crampton could only palm it into his own net.
Nyarko guided a header just wide as Bournemouth sought to restore their two-goal lead, before a Tonks cross was comfortably caught by Cisse.
Gannon-Doak went close as he latched onto a great ball in behind by Mottoh, cut inside and curled an effort towards goal from the top of the box, but the ball struck the post and bounced to safety.
Nyarko shot wide after a Gannon-Doak cross wasn’t cleared, before the Scot had an effort blocked from the right of the area.
The winger went close again with two minutes to go of the first half as he again picked the ball up from Mottoh and cut inside, but his effort went just wide of the post.
Zac Christie (Image: Robin Jones/AFC Bournemouth)
West Brom drew themselves level in the 44th minute. A throw was quickly taken in their half, Sule and Remar McNeil exchanged passes to see the former player through on goal and he finished well past Crampton.
Two minutes into the second half, the visitors took the lead.
Sule completed his brace as a corner wasn’t dealt with and he was on hand to bundle the ball over the line from close range.
The Scottish forward sealed his hattrick in the tenth minute of the half when he finished neatly with a low shot into the bottom corner across Crampton.
McNeil shot straight at the Bournemouth keeper, before Charlie Blackshields tested Crampton with a curling effort from range.
Gannon-Doak had a powerful effort caught by Cisse after he brought down an Alfie Merritt ball from halfway on the right of the Baggies’ area.
The winger showed excellent feet to create space on the right and lift a ball in for Nyarko that was well saved by Cisse.
Koby Mottoh (Image: Robin Jones/AFC Bournemouth)
That was Gannon-Doak’s final action as he was replaced by Ethan Sills in the 65th minute.
Bournemouth narrowed the gap to a single goal with 22 minutes left to play as Nyarko picked off a loose pass at the back, he played it to Mottoh who in turn found Sills.
The substitute turned the ball onto Rees-Dottin and he finished well to make it 4-3.
Crampton saved well at the near post from a Sule shot, before the keeper tipped a Deago Nelson shot onto the crossbar.
But any hopes of a Cherries comeback were dashed ten minutes from time as Dupont headed home froma Maughan corner which Crampton wasn’t able to claim.
Michael Dacosta Gonzalez doubled the Cherries lead in the first half. (Image: Robin Jones/AFC Bournemouth)
Dacosta Gonzalez drew a save out of Cisse before Rees-Dottin had a penalty shout waved away.
Merritt was replaced by Dane Ndimukum with three minutes left to play, and he was involved immediately as his powerful effort after a counterattack was parried out to Mottoh, but he could only fire over the bar.
Sule tested Crampton in search of his fourth goal on the night and Mottoh shot wide via a deflection.
But, after West Brom failed to clear the resulting corner, Christie saw his shot blocked out to Mottoh who crossed to Ameer Ali, and his header found Nyarko who applied the finishing touch to give Bournemouth hope.
The Cherries sought a dramatic late equaliser to send the tie to extra time, and Crampton came up for a late corner, but it was dealt with and the final whistle went.
That brought an end to the Cherries’ record-breaking Premier League Cup run at the quarterfinal stage.