The act have been the frontrunners since Russia's invasion - which prompted organisers to ban the country from competing.
Ukraine joined the contest in 2003 and its three wins make it one of the most successful of the newer competitor countries - having triumphed in 2004 and 2016.
During their performance, which combined rap and Ukrainian folklore, the group were dressed in elaborate outfits, including long multi-coloured fringed ensembles.
"Next year Ukraine will host Eurovision!
"For the third time in its history. And I believe - not for the last time."
Among those offering messages of congratulations, former Doctor Who star John Barrowman said: "Congratulations #Ukraine on your #Eurovision Win!
"Stick that where the sun don't shine #VladimirPutin."
The UK's Sam Ryder came in second place, and won top marks from the judges, with his uplifting pop song Space Man, marking the first time in 20 years that the UK has placed in the top three.
Graham Norton, who presented the contest on BBC One, said: "This is a red letter moment. I am so happy for him, for the UK and for the BBC who have worked so hard to turn our fortunes."
The 32-year-old Tik Tok star won over the audience, dazzling in a one piece suit encrusted with beads and pearls.
After topping the national jury vote with 283 points, beating out favourites Spain and Sweden, Ryder said: "There is so much gratitude, what an experience."
Ryder was the 22nd act to perform during the grand final in Turin, Italy, playing a guitar mid-show in front of a crowd of 7,000 fans at the Pala Olympica arena who sang along.
He found fame covering songs on TikTok during lockdown, amassing 12 million followers and catching the attention of global stars including Justin Bieber and Alicia Keys.