Watched by Klitschko's elder brother and WBC champion Vitaly, Povetkin floored the lanky Pole with a right hook in round two, then sent his opponent to the canvas again in the following round before finishing the job a few seconds later.
The 33-year-old Russian improved his record to 26-0, including 18 by knockout, while Wawrzyk suffered his first defeat in 28 professional fights.
"It was a good workout. I didn't aim for an early knockout, it just happened," Povetkin told Russian television.
Asked if he was now ready to fight Ukrainian Klitschko, Povetkin said: "Yes, let's do it."
Povetkin's manager Vladimir Khryunov added: "The Klitschko fight will be held here in Moscow, probably in early October."
Russia's WBA cruiserweight champion Denis Lebedev takes on former title holder Guillermo Jones of Panama later on Friday. (Reporting by Gennady Fyodorov, editing by Pritha Sarkar)
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