The "Gypsy King" returns this month to take on Deontay Wilder in a trilogy showdown with his WBC title on the line.
His February 2020 win over Wilder made him the only fighter after Riddick Bowe to win all four major titles at heavyweight.
Fury won the other three belts (WBA, IBF and WBO) against Klitschko in Germany via unanimous decision.
Klitschko dominated the division for more than a decade and is regarded as one of the greatest fighters of the modern era.
He retired with back-to-back defeats to Fury and Anthony Joshua, the latter beating him at Wembley.
Wladimir was left to taste another loss on Saturday night as England beat Ukraine 4-0 in Rome.
England booked their place in the semi-finals thanks to a Harry Kane brace, Harry Maguire and Jordan Henderson. They are now favourites to progress and win the tournament.
Prior to the meeting, Klitschko had wished both teams good luck while his older brother Vitali - Mayor of Kiev - featured in BBC Sport's pre-match coverage before the game.
"I've not seen a beat down over Ukraine like this since Dusseldorf 28 November 2015," Fury tweeted.
Fury is looking to make the first successful defence of his world title against Wilder and get closer to boxing history.
The 32-year-old had agreed to face Joshua in an all-British showdown to determine heavyweight's first undisputed champion. August 14 was the date and a site fee had been agreed in Saudi Arabia.
A court ruling would block the fight though, forcing a trilogy contest between Fury and Wilder, which takes place on July 24.
Joshua, who looks set to face Oleksandr Usyk, said Fury had let everyone down.
"Let me get past Usyk first. With or without Usyk in my life I'm going to fight Fury. Let's put it that way. Usyk's not the end all and be all. Usyk doesn't determine that fight," Joshua told Sky Sports.
"Unfortunately [Fury's] team let the whole boxing world down. I will still be here, still ready to put on a show.
"That fight has to happen. It's a big fight - bigger than boxing, bigger than the belts. So yeah, it'll happen.
"After I fight Usyk and defend my belts and hopefully the world's back to normal, this fight is going to be absolutely bigger and better than what it would have been in a few weeks."
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