SuÃ¡rez scored twice in the second half to reverse Stokeâ€™s advantage after Kenwyne Jones had given the home side the lead after 43 minutes.
Liverpool started brightly, showing their intent to make inroads in the competition as they bid to end their six-year trophy drought.
Andy Carroll, making only his fifth start of the season, was the first to test Stoke goalkeeper Thomas Sorensen, who blocked his low shot after a surging run by the England international.
Kenny Dalglish had opted to start Liverpoolâ€™s star man SuÃ¡rez and the forward was soon involved in the action – his cheeky chip was comfortably saved by Sorensen.
And the Uruguayan was denied again when a slick Liverpool move saw SuÃ¡rez burst past the home sideâ€™s defence – but his tame shot was pushed wide for a corner. From the resulting set-piece Carroll saw his header cleared off the line.
Meanwhile, Stoke struggled to break down Liverpoolâ€™s defence at the other end and their first real opportunity came when a trademark Rory Delap long throw caused confusion, and Jonathan Walters nodded into the net.
But the goal was disallowed as Stoke captain Ryan Shawcross was adjudged to have fouled Pepe Reina in the Liverpool goal.
After 43 minutes, against the run of play, Stoke broke the deadlock.
Liverpool summer signing SebastiÃ¡n Coates failed to clear a simple ball, allowing Walters to steal possession and cross for Jones, who managed to hold off Jamie Carragher and send a thumping header into the net just before the break.
But the home side’s lead did not last long, and seven minutes after the interval parity was restored by Liverpool’s danger-man.
SuÃ¡rez cunningly pushed the ball through the legs of Ryan Shotton and curled a sublime strike into the top corner.
The goal spurred Liverpool on and with 10 minutes remaining, substitute Craig Bellamy combined with Carroll to create space and fire a low shot onto the post.
But fittingly, it was SuÃ¡rez who had the decisive say on the outcome of the tie in the 85th minute.
Liverpoolâ€™s No7 was first to pounce on Jordan Hendersonâ€™s lofted cross and the striker made no mistake with his finish, expertly placing his header past Sorensen to seal the victory.
“We were good as well as patient. We had six chances in the first half and we came in one down.
“Great credit to the boys that they kept the belief, they kept at it and a fantastic goal for Luis followed on with a smashing header as well.
“I think you come here you know you’re always going to have to defend and if you don’t match them you’re going to be in big trouble.
“And I think the boys matched them really well and they tried to play football when they got the opportunity and they did it. We’re very pleased with the result.”
Stoke City: Thomas SÃ¸rensen, Robert Huth, Jonathan Woodgate, Ryan Shotton (Jermaine Pennant, 60), Ryan Shawcross, Matthew Etherington (Cameron Jerome, 64), Marc Wilson, Glenn Whelan, Rory Delap, Kenwyne Jones (Peter Crouch, 88), Jonathan Walters.
Liverpool: JosÃ© Reina, Martin Kelly, Jamie Carragher (Martin Skrtel, 46), Daniel Agger, SebastiÃ¡n Coates, Maxi RodrÃguez (Craig Bellamy, 82), Jordan Henderson, Jay Spearing, Lucas Leiva, Luis SuÃ¡rez (Dirk Kuyt, 88), Andy Carroll.
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