Luis Suarez capped a superb individual display with a late winner as Liverpool came from behind to beat Stoke in the Carling Cup fourth round.
The hosts went ahead just before the break when Kenwyne Jones headed a low cross from Jon Walters into the corner.
Liverpool hit back when Suarez nutmegged Ryan Shotton and curled a stunning strike past Thomas Sorensen.
And four minutes from time, Jordan Henderson volleyed a cross to the far post where Suarez headed home.
Liverpool then withstood a late onslaught to reach the last eight for the first time in four seasons and exact revenge for their 1-0 league loss to Stoke in September.
For the hosts, who had a Walters goal harshly ruled out in the first half, defeat at least enables them to focus their attention on the league and on continuing their impressive progress in Europe.
After a relatively even opening, in which Walters and Carroll both went close from long range, the Reds took control of the first half and almost went in front through Suarez.
Lucas played a one-two with Maxi Rodriguez and squared the ball for the striker, whose shot from point-blank range was smothered by Sorensen.
A fortunate deflection then fell at the feet of Carroll, but his shot with his weaker right foot was straight at the Stoke goalkeeper.
Stoke thought they had taken the lead against the run of play when Walters headed a Rory Delap long throw into the net only for referee Lee Probert to disallow the goal for a push on goalkeeper Pepe Reina.
Any sense of grievance was short-lived as Stoke took the lead following a sloppy error from Liverpool centre-back Sebastian Coates.
The Uruguayan's mis-hit clearance enabled Walters to escape down the right wing and send over a low cross for Jones to glance a header into the far corner.
Shortly afterwards, Suarez wriggled into the box and stayed on his feet under a challenge from Ryan Shawcross before dragging a shot well wide.
Stoke began the second half on the front foot but were undone by a moment of magic from Suarez.
Picking up the ball on the left, he skipped past Shotton, advanced into the area and whipped an unstoppable shot just inside the far post.
With the game drifting towards extra-time, Reds boss Kenny Dalglish introduced Craig Bellamy and the Welsh striker was soon into the thick of the action, exchanging passes with the disappointing Carroll before hitting the post from the edge of the box.
Liverpool's winner arrived moments later when a headed clearance fell to Henderson just outside the box.
The midfielder's first-time cross picked out the unmarked Suarez at the far post and the Uruguayan planted a header across Sorensen.
Suarez hobbled off soon afterwards with a calf problem and watched from the sidelines as some desperate defending prevented Stoke from taking the game into extra-time.