Justin Kluivert netted the only goal in a first half otherwise dominated by Vincent Kompany's team, who had a Josh Cullen effort disallowed in the second period before Antoine Semenyo gave the Cherries some breathing space.
As expected, Cullen was brought into the Clarets' midfield in place of the suspended Josh Brownhill, while there was a sight for sore eyes in the Bournemouth XI, as Dominic Solanke was passed fit to start after missing the FA Cup loss to Leicester City with a knee issue.
Kompany's men - the worst home team in the Premier League this season - made the quicker start at Turf Moor and forced Neto into the first save of the match after just four minutes, but Wilson Odobert's first-time strike from a central area was too tame to trouble the Brazilian.
The Cherries weathered the early storm, but Andoni Iraola was dealt another injury blow in the 12th minute as Marcos Senesi was withdrawn for Chris Mepham; the defender pointed to the back of his thigh after taking his place on the bench.
However, just one minute after that unplanned alteration, Iraola's troops drew first blood through Kluivert, who raced through to meet Lewis Cook's ball over the top, gave Dara O'Shea the slip and lashed in a half-volley.
After dominating the early moments only to fall behind - a familiar sight for the Turf Moor faithful - Burnley began to force the issue once more, but a distinct lack of ruthlessness was frequently hampering the hosts.
Vitinho - playing in a more advanced role - curled over the bar from inside the box in the 22nd minute, a fate shared by Odobert with a 35-minute long-range strike which just missed the upright.
The latter then slipped in David Datro Fofana with a slick through ball in the 41st minute, but Neto did enough to thwart the Chelsea loanee, and Jacob Bruun Larsen's follow-up was blocked by the thigh of Adam Smith.
Kompany's men were getting closer, though, and Bruun Larsen was only an inch or two away from equalising with a sweetly-struck 45th-minute free kick, which partially crossed the line before Neto acrobatically sprung across his goal to make a crucial save.
Another promising opening passed the hosts by in the first additional minute, as Vitinho nodded over from Charlie Taylor's inch-perfect delivery, and while his side survived until half time, Iraola bit the bullet and brought on Alex Scott for Ryan Christie to re-energise his midfield.
The Cherries almost punished Burnley for their first-half profligacy in the 56th minute, but James Trafford was equal to Solanke's strike from a tight angle, before Marcus Tavernier made a smart run through the middle and found the side netting one moment later.
The ball finally found its way into the Bournemouth net in the 66th minute, as Cullen hooked a close-range attempt into the corner after the visitors failed to deal with a cross, but David Coote penalised Bruun Larsen for a foul on Smith and the goal was chalked off.
Following that reprieve, Bournemouth attacker Semenyo wasted two gilt-edged chances to kill the game off in the 75th and 76th minutes, firing wide of Trafford's far post on both occasions after working a yard of space in the right-hand side of the box.
However, the 24-year-old made no mistake with his left foot in the 88th minute, nonchalantly advancing into the box from the right, cutting inside and finding the top corner - with the help of a slight deflection - to rubber-stamp Bournemouth's first Premier League triumph of the calendar year.
Sunday's result has no impact on the Cherries' 13th-placed ranking or Burnley's 19th-placed standing in the table, although Iraola's side are now 11 points clear of the drop zone, while Kompany's team remain 11 points from safety.
A clash of the Clarets is next on the menu for Burnley next Sunday, travelling to the London Stadium for a date with West Ham United, one day after Bournemouth host Sheffield United.