Darwin Nunez has scored seven goals across all competitions this season
The Uruguayan striker cut in from the left and picked his spot perfectly for the 70th-minute finish, curling the ball with plenty of pace into the top right corner beyond a despairing Ionut Radu.
Cody Gakpo had put Liverpool in front in the first half when he pounced to poke home from close range after Harvey Elliott's volley was parried into his path.
But Bournemouth levelled through Justin Kluivert as he netted his first goal for the club since joining from Roma in the summer. He emerged unmarked at the far post to head home an Alex Scott corner in the 64th minute.
The hosts committed everything to attack during the final 15 minutes. David Brooks and Hamed Traore chanced their arm, but neither could muster an attempt as the Cherries failed to level for a second time.
Liverpool will take on Premier League rivals West Ham, who beat Arsenal 3-1 in the fourth round, in the quarter-finals at Anfield.
Bournemouth are still getting to grips with the philosophy of manager Andoni Iraola, who replaced Gary O'Neil over the summer.
Saturday's 2-1 win against newly promoted Burnley - their first victory of the campaign - was badly needed to move them one point clear of the bottom three.
As enjoyable as a run in the EFL Cup can be, Bournemouth cannot afford to get distracted from the main priority of retaining their spot in the top-flight for a third successive season.
The road ahead may still be long and arduous for the Cherries but there were some positives to take from their loss to Liverpool.
With the wind and rain swirling around the Vitality Stadium as Storm Ciaran hit, Bournemouth went toe-to-toe with their visitors in the opening 45 minutes - each side registered four shots on target.
Kluivert was responsible for two of those efforts and the Dutch forward will have been pleased to finally get on the scoresheet in Bournemouth colours.
Midfielder Scott, another summer arrival from Bristol City, also caught the eye and he was inches away from scoring directly from a corner as he used the conditions to whip in a ball that Joe Gomez had to clear off the line.
Chants of "Nunez, Nunez, Nunez" rang out around the Vitality when the full-time whistle went.
The 24-year-old struggled to live up to his £64m price tag last season, scoring 15 goals across all competitions, but he has fitted well in a Liverpool shirt in this campaign.
With Jurgen Klopp opting to rotate his starting 11 - just three players survived from Sunday's 3-0 win against Nottingham Forest - Nunez had to make do with a place on the bench.
Brought on in the second half with the game level at 1-1, the frontman took just nine minutes to put his stamp on the match, bending a beautiful effort across goal to register for the third successive game and take his haul to seven this season.
It was an unusually quiet night for Mohamed Salah, one of those three to retain their starting berth and the only player in the Liverpool ranks to have scored more than Nunez in 2023-24.
The Egyptian had one sight of goal in the second half but could only head wide.
Liverpool's summer revamp clearly seems to have had the desired impact as the win at Bournemouth extended their unbeaten streak to six games, while they have lost just once all season.
-- Courtesy of BBC Sport