The win in the first league game of the weekend sees the Cherries rise from last spot to 16th in the table on 24 points, while Liverpool languish in fifth on 42 points after wasting a chance to close the three-point gap to fourth-placed Tottenham Hotspur.

Liverpool started well, with Mohamed Salah scuffing an early shot goalwards and Cody Gakpo having an effort ruled out for offside.

Bournemouth defender Jefferson Lerma also had to clear a Virgil van Dijk header off the line, but as so often before this season, the wheels quickly came off for the visitors.

Bournemouth's Dango Outtara fired wide after a one-on-one with Alisson but the Reds didn't heed that warning, and they paid for it in the 28th minute when he outpaced Van Dijk and squared the ball for Billing, who steered it home.

Former Liverpool player Dominic Solanke almost added a second just before the break and Billing went sprawling in the box, but his appeals for a penalty were waved away as Bournemouth closed out the half strongly.

Diogo Jota came on for Harvey Elliot at halftime and quickly made his presence felt, forcing a save from Neto with a stinging shot four minutes after the break.

Jota then won a penalty for Liverpool as his goal-bound header struck the arm of Adam Smith, but to the horror of the travelling fans Salah blasted his spot kick high and wide of the goal.

That miss in the 70th minute seemed to deflate Liverpool entirely, and they did precious little to threaten the Bournemouth goal, with Gakpo flashing a late opportune effort over the bar as his side's attack came up short again.

The home fans cheered every tackle and clearance and they erupted in joy at the final whistle as their side put some daylight between themselves and the rest of the clubs at the bottom of the table.

"(We had to) be diligent, be compact and work for each other, as we've done loads of times this season. Today it paid off and the support was brilliant as well," jubilant goal-scorer Billing told broadcaster BT Sport.

Liverpool manager Juergen Klopp was left to pick over the remains of another poor performance from his side in a season that has never really got going for them.

"The spaces where we wanted to play were super-clear, they were super-open ... we didn't use that often enough, we didn't enjoy the challenge of facing a deep, compact side," Klopp said, and he declined to blame the defeat on Salah's penalty miss.

"If you score there, it doesn't make the performance better but it could change the result. He (Salah) scores a lot of goals, but he missed the penalty. That's life," Klopp said.