The Cherries’ FA Cup campaign came to an end in the fourth round with a 2-0 defeat against Liverpool.
Luis Suarez created both goals, which were scored by Victor Moses and Daniel Sturridge either side of half-time.
The home side created a number of chances early on with Andrew Surman and Marc Pugh heading just off target, while Simon Francis had a long range free-kick tipped over.
But Moses found the back of the net in the 25th minute, cutting in from the left and powering a shot past Lee Camp at the near post.
England international Sturridge sealed the win with a classy finish on the hour mark following an incisive pass from his Uruguayan strike partner.
Playing a fourth round tie on the south coast for the first time in 50 years, the Cherries looked to set up an uncomfortable end for the Reds but were unable to pull a goal back, although Brad Jones did push a swerving strike from Lewis Grabban over.
Roared on by the home crowd, the Cherries made a positive start and both Pugh and Surman saw headers narrowly miss the target.
Liverpool withstood the early pressure before captain Steven Gerrard fired a free-kick just over the angle of post and bar.
The Reds then took the lead in the 25th minute as Suarez picked out the unmarked Moses, who cut inside and found a way past Camp.
Eddie Howe’s team were far from outclassed and some of the home supporters were out of their seats when Harry Arter tried his luck from the edge of the area, but the ball flew just wide of the target.
The Cherries kept up the pressure and Kolo Toure needed to put in a brave block to deny Surman, who then crossed for Tommy Elphick to head just over.
However, the visitors should have been two up heading into the break, but Jordan Henderson guided his effort from close range over having been set-up by Suarez.
Suarez then turned provider on the hour mark as he played in Sturride, who extended Liverpool’s with a calm finish in front of the visiting supporters.
Sturridge almost added a third, seeing his lob from the edge of the area bounce off the top of the crossbar as the Reds’ Premier League began to show.
Howe introduced Ryan Fraser, Tokelo Rantie and Brett Pitman from the bench and they gave the home side a lift heading into the final 20 minutes.
Rantie’s pace caused some problems as the Cherries looked for a way back into the game and to set up an uncomfortable final few minutes for the seven-time winners.
Camp was called upon once more in injury time to save Suarez’s near post effort after Grabban’s swerving strike from distance had been beaten away by goalkeeper Jones.
But Brendan Rodgers’ team were able to see the game out and look forward to seeing their names in Sunday’s fifth round draw.
Cherries: Camp, Francis, Ward, Elphick, Daniels, Ritchie (Fraser 72), Arter (Pitman 78), Surman, O’Kane, Pugh (Rantie 72), Grabban.
Unused substitutes: Allsop, Cook, Harte, MacDonald.
Liverpool: Jones, Kelly (Flanagan 73), Toure, Skrtel, Cissokho, Moses (Sterling 83) Gerrard, Henderson, Coutinho (Alberto 83), Sturridge, Suarez.
Unused substitutes: Mignolet, Ibe, Sama, Aspas.
Goals: Moses (25), Sturridge (60)
Attendance: 11,475 (including 2,277 Liverpool supporters)