The Merseyside based outfit made a positive start to the game, seeing a shot cleared off the line and a goal disallowed for offside before the Cherries broke the deadlock from a quick counter.
Ouattara got them upfield and Billing converted by slotting past Alisson Becker.
The home side defended with everything they had in the second period but Liverpool were given a route back into it when Smith was penalised for a handball in the 18 yard area.
Egyptian winger, Mohamed Salah sent the resulting penalty well off target though, and they couldn’t find a way back.
That victory lifts Bournemouth off the foot of the table and into 16th place for now.
They get their first home win since the restart and look ahead to their game at Aston Villa next weekend. Liverpool miss the chance to go fourth as their unbeaten run in the league comes to an end.
They travel to Real Madrid in midweek to try and overturn the deficit in the Champions League round of 16 second-leg tie.
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