Burnley manager Sean Dyche praised his side who overcame a jittery start at the Vitality stadium.
"So many parts of the performance were strong today," Dyche told the BBC. "There was no nervousness and we stayed clear-minded. There was a real calmness to our play.
"Twenty-four points since Christmas is a very strong return. We kept playing, we kept probing, we kept doing all the diligent things that we do well.
"It puts us in a much healthier situation than we were two games ago."
The home side took a fourth-minute lead after Barnes headed into his own net but Burnley came roaring back as Chris Wood levelled with an 18th-minute header before Ashley Westwood swept in a shot from 13 metres barely two minutes later.
Both goals resulted from sloppy Bournemouth defending as their keeper Asmir Begovic misjudged an inswinging corner for the equaliser before Chris Mepham's poor clearance gave Westwood time and space to fire the visitors ahead.
Callum Wilson missed a gilt-edged chance to equalise midway through the first half when he headed over the bar from close range before another Begovic howler allowed Barnes to score at the right end in the 57th minute.
Wood beat the indecisive Begovic to a deflected cross at the near post, as he turned swiftly and squared the ball back to his strike partner Barnes who rifled it confidently into an empty net from eight metres.
Bournemouth manager Eddie Howe was unhappy with his team's bloodless performance which stretched their winless home run to four games, leaving them with only one win from their last nine league outings.
"There were lots of elements from our performance today which wasn't great," said Howe.
"I have always backed our goalkeepers. Mistakes happen in football. Today we made numerous mistakes in areas we wouldn't expect them to happen.
"We pride ourselves on our professionalism and our ability to never give up but it didn't look like that today. We are on a difficult run and confidence is maybe a bit low but the season isn't over yet."
(Writing by Zoran Milosavljevic; Editing by Pritha Sarkar)
