Bournemouth started well and had a superb chance early on as Surman headed for goal but saw his effort go over the bar, Arter hit a threatening pass across the area which was cleared by Kelly and Francis tested Jones with a clever free-kick.
Liverpool began to get into the game with Moses and Suarez both creating space but it was minutes later that they made an impact as Suarez picked out Moses who fired past Camp to open the scoring and give the visiting fans some joy after 26 minutes.
Bournemouth fought back and Surman forced a vital block from Toure with Elphick heading over from a dangerous corner, both sides pushed for a goal as Suarez capitalised on a defensive mistake to pick out Henderson who hit over the bar.
Skrtel ensured that the lead stayed with Liverpool by denying Arter who's effort looked set to beat Jones as the game went into the half-time break. Bournemouth continued where they had left off by pegging Liverpool back which saw a tactical switch of Gerrard dropping to centre back while Sktrel had to be treated after a collision with Surman.
The change didn't phase Liverpool though as they increased the lead on the hour-mark when Gerrard found Coutinho who in turn found Suarez and the Uruguayan picked out his fellow SAS partner Sturridge who hit past Camp for a 2-0 lead.
Bournemouth could have gone further behind in the 76th minute when Sturridge returned the favour to Suarez but the resulting lobbed effort went over the bar and Liverpool held on for a safe passage to the fifth round which will be drawn tomorrow.