Mohamed Salah's glaring penalty miss helped Bournemouth climb out of the relegation zone after an excellent win which damaged Liverpool's chances of qualifying for next season's Champions League.
The Reds were brought back down to earth after last week's 7-0 thrashing of Manchester United as the Cherries, who started the weekend bottom, did their survival chances a power of good.
Liverpool were punished for poor defending when the unmarked Philip Billing struck from eight yards for his sixth Premier League goal of the season after fine work by Dango Ouattara.
Virgil van Dijk had a header cleared off the line at 0-0 and substitute Diogo Jota was denied by Neto's fine save before Liverpool won a penalty when the Portugal forward's header was handled by Adam Smith.