The Reds came from behind against the Potters to move back above City into second and six points behind Premier League leaders Chelsea.
Jon Walters headed the Potters in front but goals from Adam Lallana, Roberto Firmino, an own goal from Giannelli Imbula and a Daniel Sturridge strike turned it around for the hosts at Anfield.
City boss Guardiola watched from the stands ahead of their New Year’s Eve clash, but Kop boss Klopp believes his side can do better.
He said: “It is a big game. It is a big game for both. I heard Pep Guardiola was here in the stadium tonight.
“I’m not sure if he watched a lot of games in the last few weeks in other stadiums of opponents.
“That is a first sign it is a special game and we are already looking forward to it.
“A lot of things we could have done better, but we forced two goals with our quality.
“Hopefully we can do better (against City) because we can do better than we did tonight but tonight it was absolutely OK.”
This scoreline would not have been predicted by anyone after the first 25 minutes as Liverpool looked like they had over-indulged at Christmas.
Liverpool’s former midfielder Joe Allen was running the show for Stoke early as boss Mark Hughes went in search of his first ever win at Anfield as a manager.
It looked possible at the 11th attempt when Jon Walters headed in after just 12 minutes as the boyhood Everton fan rose to nod Erik Pieters’ cross home after some woeful defending from the hosts.
Liverpool’s sloppy play continued as they gave possession away and Allen forced a good stop from Kop keeper Simon Mignolet that could have changed the course of the game.
But then they got their own gift in the 35th minute from another familiar face at Anfield.
Lallana broke into the area and accidentally played a one-two with Stoke defender and former Red Glen Johnson before slotting home at the near post.
The mood changed at Anfield and suddenly the Kop was roaring the Reds on.
Hughes was desperately trying to motivate his side again as he stood on the touchline barking out orders.
But the Potters only had themselves to blame for Firmino’s strike just before half-time.
The Brazil ace, who was arrested on Christmas Eve for drink driving, was given plenty of breathing space in the area by Johnson and Ryan Shawcross before he found the net via both posts.
Liverpool had not lost their last 15 Premier League home games before this and they didn’t look like that run would be broken as soon as they got the second.
And the third present arrived on the hour mark for the hosts when Imbula turned into his own net after Jordan Henderson had found Divock Origi to cross.
With the three points in the bag, Klopp sent on Emre Can and Sturridge to get some minutes.
Sturridge needed only 56 seconds to get his first league goal since April thanks to the biggest present of the day from Shawcross.
The Stoke defender played a blind pass back to keeper Lee Grant and the England striker was on it in a flash.
Sturridge used his first touch of the game to round Grant and then his second to find the net.
Stoke boss Hughes said: “Liverpool are a good side, they have got good players and will go close – but whether they’re good enough to win it I’m not sure.”
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