The Cherries went ahead early on through Joshua King before Dan Gosling then doubled the advantage shortly after.
But the hosts fired back through Andre Ayew and Jonjo Shelvey to level proceedings before half time.
It was the Cherries that started the better of the two sides, with Junior Stanislas looking threatening early on. A smart turn on the edge of the penalty area allowed the 25-year-old to tee up the onrushing Gosling, but the midfielder's shot flew just over the Swansea crossbar.
AFC Bournemouth then took the lead on the 10th minute through King. Stanislas capitalised on a defensive error to break free into the Swans penalty area, before crossing for the unmarked King to slide past Lukasz Fabianski.
Five minutes later King had a chance to double the advantage. Picking up a loose ball on the halfway line the Norwegian international bore down on the Swansea goal, but Fabianski pushed the forward's strike behind for a corner.
The Cherries soon had the ball in the net once more. A deep cross from Simon Francis allowed Stanislas to volley in off the post., but the celebrations were cut short by the linesman's offside flag.
There was to be a second on the 15th minute, though. Charlie Daniels' surging run resulted in Matt Ritchie crossing for Gosling, who then tucked the ball into the top corner of the Swansea net.
The hosts were quick to react and soon snatched one back. A half cleared cross found its way back into the Cherries penalty to Ayew, who backheeled past Federici from close range, despite protests for offside.
At 2-1 Swansea appeared to spring back into action. A frantic couple of minutes of end to end action followed, as both sides looked to take control of the encounter.
It was to be the hosts that drew next blood, though. A tangle between Francis and Ayew resulted in Andre Marriner pointing to the spot, allowing Shelvey to step up and level proceedings before half time.
The Cherries started brightly after the interval. A Harry Arter strike from distance tested Fabianski early on and Steve Cook's volley from a half-cleared corner whistled just past the Swansea post.
Eddie Howe then looked to shake things up in the closing minutes, as Adam Smith and Glenn Murray replaced Ritchie and King.
Murray almost made an immediate impact as he hooked the ball into the Swansea penalty area, but the tireless Gosling was unable to prevent his strike down from sailing over the hosts' crossbar.
Not to be outdone, Arter then burst down to the byline and delivered a cross to the near post for Murray to tuck home, but the summer signing was unable to steer his effort past the outstretched Fabianski.
Cherries: Federici, Daniels, Distin, Cook, Francis, Gosling, Surman, Arter (MacDonald, 90), Stanislas, Ritchie (Smith, 79), King
Not used: Allsop, Kermorgant, Murray, O’Kane, Cargill