A mouth-watering Premier League Cup qualifying round saw Michael Jolley's side progress into the last 32 after a 3-2 extra-time thriller at Turf Moor.
Freddy Yao bagged the winner in the second period of the additional 30 minutes after goals from Dwight McNeil and Tinashe Chakwana has earlier twice cancelled out Cherries’ leads.
It was the visitors who started on the front foot in the first 10 minutes - and got their reward and Mikael Ndjoli played a neat through ball to Jordan Lee.
The full back squared the ball into the box and after Ben Whitfield was denied by Conor Mitchell, Sam Surridge followed and slotted home.
Burnley looked to reply minutes later as Mark Howarth's corner fell to winger Bradley Jackson just outside the box, who drilled an effort at Callum Stanton in the Bournemouth goal.
However, the home side's search for a leveller came as McNeil picked up the ball outside the area and curled an effort into the back of the net past Stanton.
The Cherries were not behind for long though and on the half hour, Ndjoli's driving run at Jimmy Dunne paid dividends as the winger fired past Mitchell in the far corner.
The absorbing tie took another twist in the second half when Chakwana turned just outside the box and drilled home past Stanton.
Chakwana scores the Clarets' second.
And as Michael Jolley's side took control they almost grabbed a third when Taofiq Olomowewe played a neat one two with Yao and fired just wide.
Bournemouth also came close to scoring a winner late on, as substitute Matt Worthington saw his effort denied from Mitchell.
Extra time was a more sedate affair until a Tom Anderson header fizzed past the post.
However, Burnley got their rewards three minutes into the second extra half as substitute Yao was brought down in the box by Lee.
Yao takes the plaudits
And the former Stoke City scholar followed up his saved penalty to smash home the decider.
BURNLEY: Mitchell, Olomewewe, Dunne, Anderson, Birch (Yao 43), McNeil (Pingling 62), Howarth, Dummigan, Jackson, Chakwana, Massanka (Bayode 90).
Subs: Aghayere, Leitch