Sean Dyche’s side made the worst possible start at Dean Court when Barnes headed into his own net as he tried to clear under pressure from Bournemouth’s Nathan Ake in the fourth minute.
However, Burnley were gifted an equaliser in the 18th minute as Bournemouth keeper Asmir Begovic let Ashley Westwood’s corner go straight through his hands and Chris Wood nodded in at the far post.
Inspired by the New Zealand forward’s 11th goal this season, the Clarets struck again two minutes later, with Westwood firing home after Chris Mepham failed to clear Dwight McNeil’s cross.
Burnley were well on top and Barnes struck at the right end in the 57th minute.
Once again Begovic was guilty of poor handling, allowing the ball to slip to Wood, who laid it back for Barnes to slot into the empty net.
Burnley are eight points clear of the relegation zone, and although third bottom Cardiff have a game in hand, Dyche’s men have taken a major step towards staying up.
Leicester powered to a 4-1 rout of relegated Huddersfield as Brendan Rodgers’ side moved into seventh.
Youri Tielemans put Leicester ahead at the John Smith’s Stadium with his second goal since joining on loan from Monaco.
The Belgian midfielder netted with a deflected strike from the edge of the area after Huddersfield were unable to clear Jamie Vardy’s miscued shot in the 24th minute.
Vardy doubled Leicester’s lead three minutes after half-time, collecting Ricardo Pereira’s cross and slotting home after rounding keeper Ben Hamer.
Huddersfield midfielder Aaron Mooy reduced the deficit with a 52nd minute penalty controversially awarded for Caglar Soyuncu’s challenge on Karlan Grant.
James Maddison put the result beyond doubt with a fine free-kick in the 79th minute.
Bottom of the table Huddersfield, playing their first home match since relegation was confirmed, endured one last blow when Vardy stroked in an 84th minute penalty after the England striker was brought down by Terrence Kongolo.
Luka Milivojevic extended his superb penalty record as Crystal Palace won 1-0 at Newcastle to climb 11 points clear of the bottom three.
Newcastle defender Florian Lejeune was stretchered off with his left knee heavily strapped after a nasty second half collision with Andros Townsend that left him needing a hospital scan.
Roy Hodgson’s team took the points in the 81st minute when Wilfried Zaha was clipped by DeAndre Yedlin and Serbian midfielder Milivojevic slammed in his 10th successful spot-kick this term.
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