Charlie Taylor sent Burnley ahead in the 11th minute of the contest, but Antoine Semenyo levelled the scores in the 22nd minute, before Billing's long-range effort sent the hosts ahead in the 76th minute.
The result has moved Bournemouth out of the relegation zone into 17th position in the table, with the Cherries finally recording their first league win of the season, while Burnley have dropped to 19th, boasting just four points from their opening 10 matches of the campaign.
Zeki Amdouni headed over the Bournemouth crossbar early in the match, with the 22-year-old meeting a delivery from the right, and it was a purposeful start from the visitors.
Burnley made the breakthrough in the 11th minute of the match through Taylor, with the defender's brilliant strike from outside the box flying past the home side's goalkeeper Ionut Radu.
Bournemouth should have levelled in the 21st minute when Marcus Tavernier found himself free inside the Burnley box, but the attacker somehow fired over the crossbar.
The Cherries did manage to score less than one minute later, though, with Semenyo breaking into the Burnley box before picking out the bottom corner with an excellent finish.
Bournemouth were the team on the front foot as the first period headed towards its latter stages, with Semenyo's direct running causing all sorts of problems for Vincent Kompany's side.
Burnley goalkeeper James Trafford made a smart save to keep out a header from Chris Mepham late in the first period, with the centre-back meeting a corner from Tavernier.
Semenyo then had a powerful strike deflected just wide of the Burnley post in the final minute of normal time at the end of the first period, with Bournemouth finishing on the front foot, but the two teams entered the half-time interval on level terms.
Trafford made another smart save to keep out an effort from Semenyo just before the hour, before Alex Scott sent a free kick just over the Burnley crossbar in the 68th minute.
The Clarets were struggling to offer anything in the final third of the field, but they were remaining strong defensively, with the two teams still level heading towards the final 15 minutes.
That changed in the 76th minute, with Billing finding the back of the net with a wonderful effort from distance, beating Trafford, who got a hand to the ball while back-tracking but was unable to keep it out.
Burnley thought that they had levelled in the final minute through Jay Rodriguez, but his effort was chalked off for offside following an incredibly long - around five minutes - VAR check, with Bournemouth securing all three points after surviving two late efforts from Sander Berge.
The result has ended a four-game losing run in the league for Bournemouth, while it is now three straight losses for Burnley in England's top flight.
Both teams will now switch their attention to the EFL Cup, with Bournemouth hosting Liverpool in the last-16 stage of the competition on Wednesday, while Burnley will head to Everton on the same night.