A first half goal from Nathaniel Clyne was enough to separate the two sides, despite Matt Ritchie coming close in the dying moments for the Cherries.
An early break allowed Harry Arter to slip in Junior Stanislas on the left, but his drilled effort was deflected clear off Adam Bogdan.
Liverpool then started to find their stride and Roberto Firmino struck an effort from the edge of the area, but Adam Federici was equal to it.
The Brazilian then received the ball in a similar position moments later, but this time slipped in Joao Teixeira whose shot was cleared off the line by Adam Smith. But Clyne, arriving late at the back post, was able to hammer the rebound in to give the home side the lead.
The Cherries responded quickly though. Marc Pugh went on a darting run down the left and crossed for Stanislas at the near post, but his header was tipped behind by Bogdan from close quarters.
The midfielder then had the chance to net an equaliser just before the break as his 25-yard free kick almost caught Bodgan out at his near post.
Eddie Howe’s men began the second half well, and Shaun MacDonald rose well to power an early header just wide of the Liverpool post.
Firmino then had a chance to double Liverpool’s lead on the hour mark. Picking up a loose ball in the middle of the park the 24-year-old drove into the Cherries box but dragged his shot wide of Federici’s far post.
Lucas Leiva then came close with just ten minutes remaining, but Federici was able to parry the second-half substitute’s rasping strike and clear it to safety.
Federici then made another outstanding save five minutes later as he pushed Teixeira’s free-kick safely behind his near post.
The Cherries had a chance to take the game into extra time in the dying moments as Ritchie burst into the penalty area, but the Scotland international had his effort saved by Bogdan.
Cherries: Federici, Smith, Francis, Distin, Daniels, Arter (King, 69), MacDonald, Ritchie, Stanislas (Rantie, 82), Pugh, Kermorgant (Tomlin, 69)
Not used: Allsop, Cook, O’Kane, Cargill
Attendance: 41, 948