FA Cup: Bournemouth 0 Liverpool 2
Bournemouth more than matched their Premier League visitors, particularly in the first half, but Victor Moses handed Liverpool the lead in the 26th minute with their first shot on target.
Brendan Rodgers' side had to wait until the second half to get another effort on target and it doubled their lead - Daniel Sturridge scoring his 14th goal of the season in all competitions to break Bournemouth's spirited resolve.
The Championship side have not made it beyond the fourth round since 1988-89, while Liverpool put memories of being dumped out at this stage by Oldham Athletic last season behind them.
Eddie Howe's men were unchanged from their 1-1 league draw against Watford last weekend, while Rodgers made three alterations from Liverpool’s 2-2 draw with Aston Villa, bringing in Brad Jones, Moses and Martin Kelly for Simon Mignolet, Raheem Sterling and the injured Glen Johnson.
The hosts started brightly at Dean Court, and Simon Francis almost caught Jones off guard in the 13th minute when a free-kick from the left wing was carried goalwards by a coastal gust, forcing the retreating Liverpool goalkeeper to tip over his crossbar.
Liverpool's first chance of the match came in the form of a 25th-minute Steven Gerrard free-kick, but the captain drilled his effort high and wide of the target.
However, the top-flight outfit were ahead a minute later, when Moses picked up the ball at the edge of the penalty area and drove a low effort beyond the outstretched right arm of Lee Camp and into the bottom left corner of the net.
The goal was somewhat harsh on Bournemouth, who had equipped themselves more than adequately in the opening stages, but they instantly began to push for an equaliser, with Harry Arter seeing an effort drift just wide of the target on the half-hour mark, before Andrew Surman had a 12-yard strike deflected over by Kolo Toure and Tommy Elphick headed high from the resulting corner.
Sturridge had the ball in the back of the net shortly before the break, but the linesman had flagged for an extremely marginal offside.
Bournemouth continued to press at the beginning of the second period and had cause to feel aggrieved in the 58th minute, when Kelly clearly tugged on the shirt of Francis in the Liverpool penalty area.
And their hopes were dealt a killer blow on the hour mark when Luis Suarez found Sturridge with a sublime through-ball, and the England striker made no mistake in slotting beyond Camp.
Sturridge could have made it three 12 minutes later, but his chipped effort dropped onto the top of the crossbar, and Suarez was also denied from close range late on.