Liverpool set up a semi-final meeting with Chelsea as Raheem Sterling scored twice in a Capital One Cup win away to Championship leaders Bournemouth.
The makeshift striker nodded in the opener after Callum Wilson had missed a glorious chance to put the hosts ahead.
Lazar Markovic's precise finish doubled the lead, but the Reds were thankful for Yann Kermorgant's glaring miss.
Sterling stroked in after the break and, although Dan Gosling bundled in to set nerves on edge, the Reds held on.
The victory should provide a welcome shot of confidence for a Liverpool side that had won only two of their previous 10 games and face an in-form Arsenal in the Premier League on Sunday.
Sterling's brace was the first time he has found the net for his club since September and, after missing a succession of chances in the weekend's defeat to Manchester United, he looked more composed in a new central role. His return to goal-scoring form might also be timely amid reports that talks with Liverpool over a new contract have stalled.
With Liverpool desperately short of form and Bournemouth topping the Championship on the back of a 12-match unbeaten run, only ten league places separated the sides at kick-off.
But, after Wilson had stabbed wide with only goalkeeper Brad Jones to beat, the Premier League visitors made that narrow gap feel like a chasm in class.
Liverpool dominated possession and had already threatened through Adam Lallana and Steven Gerrard's crisp long-range strikes, when Sterling capped a 52-pass move with the opener.
Sporting a newly shaved head, the 20-year-old stooped to glance home a close-range header after Markovic's deep cross had been nodded into the six-yard box by Jordan Henderson.
Deployed as an advanced wing-back, Markovic had seen plenty of the ball and he played a part in setting up his second with a neat pass to Philippe Coutinho.
As Bournemouth defender Tommy Elphick turned his back, apparently distracted by a whistle from the crowd, the Brazilian fired a snap shot from a tight angle.
Goalkeeper Artur Boruc blocked his effort but the ball ran to Markovic who thumped a side-foot half-volley through a forest of legs and into the bottom corner from 15 yards out.
Bournemouth's raids forward were rare, but against a jittery defensive trio of Dejan Lovren, Kolo Toure and Martin Skrtel, always looked dangerous.
Liverpool's lead should have immediately been halved when Simon Francis burst down the right and squared for Kermorgant, but the Frenchman blazed over with the goal gaping and little pressure.
The hosts were left to rue what could have been as the game was taken away from them by Sterling after the break.
With the Bournemouth defence pushing a higher line, the England international exploited the space in behind, twisting and turning Elphick before rolling home a composed finish.
Liverpool were briefly shaken when Matt Ritchie's cross was ushered past Skrtel and Toure by former Evertonian Gosling and his shot beat an unconvincing attempt to save by Jones.
With former Crystal Palace manager Tony Pulis looking on from the BBC 5 live commentary box, Bournemouth briefly threatened to stage a repeat of the Eagles' famous comeback against the same opposition at the end of last season.
Wilson came close to turning in Ryan Fraser's looping header, before Dan Gosling smacked the upright with a curling effort.
But with substitute Emre Can shoring up midfield, Liverpool regained control to see out time and close out the game in relative comfort.
Bournemouth boss Eddie Howe: "We showed our quality, showed we can create chances against a good team, but you have to admire their quality, they have world-class players.
"We were too open, got caught between two schools.
"We tweaked things at half-time, we took a risk with two strikers but we wanted to be positive, we were at home in a cup tie and wanted take the game to Liverpool to give it a go."