Both managers named strong teams, and a crowd of just short of 25,000 at the Britannia Stadium were rewarded with a first half full of chances that ended with Kenwyne Jones heading in the opening goal after Jon Walters had robbed Sebastian Coates.
But the night was to belong to Suarez. Nine minutes into the second half the Uruguayan curled an exquisite shot into the corner before, with extra-time looming, heading in the winner five minutes from time.
Stoke were unbeaten at the Britannia Stadium this season, with Liverpool among their victims in the Barclays Premier League in September.
A penalty from Walters separated the teams that day, and the striker came close to giving the hosts an early lead here with an audacious effort from wide on the left that dropped just over the bar with Jose Reina back-pedalling.
At the other end Stoke backed off and allowed Andy Carroll a run and shot that was parried by Sorensen into the path of Suarez, but the ball came to the Uruguayan very quickly and he could only direct it wide.
It was a breathless start to the game, and Ryan Shotton should really have done better than drag his shot well wide when he found himself in space at the back post.
Liverpool were looking marginally the more threatening and Sorensen produced a smart save to deny Suarez after a neat move involving Lucas Leiva and Maxi Rodriguez, while Carroll did not make the most of a free header.
The striker had another golden chance 26 minutes in, shooting straight at Sorensen from 15 yards with Stoke appealing in vain for offside.
The Potters thought they had opened the scoring in the 35th minute when Walters flicked in Rory Delap's long throw but referee Lee Probert felt Ryan Shawcross had impeded Reina and it was disallowed.
They had the ball in the net again nine minutes later, and this time it did count, although it was a goal Liverpool will feel should have been prevented.
Uruguayan defender Coates, making only his second start for the club, was robbed by Walters out on the right and the striker charged into the area before crossing for Jones to head into the far corner.
Nine minutes into the second half Liverpool drew level with a goal of real quality from Suarez. After collecting the ball wide on the left, the Uruguayan nutmegged Shotton before curling a delightful shot into the far corner.
After the glut of chances in the first half, the second was much tighter, with Liverpool coping comfortably with a number of Delap long throws.
Dalglish made his first change in the 82nd minute, bringing on Craig Bellamy for Maxi, and the striker almost made an immediate impact, striking the post with a low effort from 20 yards.
But Liverpool were to get another chance as Jordan Henderson's 85th-minute cross picked out the head of Suarez, who nodded home the winner.
The night ended on a sour note for the striker as he hobbled off moments later to be replaced by Dirk Kuyt, but Liverpool withstood a number of late Stoke salvos to book their place in the last eight.