Burnley's slim hopes of staying up suffered another setback as Bournemouth won in the Premier League for the first time in 2024.
Justin Kluivert set the Cherries on their way with a powerful finish past James Trafford after poor defending.
Burnley responded strongly with Neto producing an outstanding save to keep out Jacob Bruun Larsen's free-kick while Vitinho headed a good chance over the bar.
Larsen also had a goalbound shot blocked by Adam Smith after Neto denied David Datro Fofana.
Josh Cullen's disallowed goal - ruled out for a foul in the build-up - only added to the sense of frustration inside Turf Moor, before Antoine Semenyo wrapped up the points with a deflected finish in the 88th minute.