Yeliseyev reportedly practised parkour, an urban challenge which involves climbing or leaping across roofs, fences or other man-made obstacles.
He became world junior chess champion in 2012 and was a grandmaster aged 17.
He won the Moscow Open 2016 chess tournament and ranked 42nd among Russian grandmasters. His world ranking was 212.
Police quoted by Gazeta.ru news website said the marks on his body pointed to his having plunged from the 12th-floor balcony on Saturday night.
"Tonight my close friend died - an outstanding chess player and analyst, one of the most talented people I know, Yura Yeliseyev," wrote Daniil Dubov on Facebook.
"He was trying to climb from the window onto a balcony on the 12th floor but lost his grip."
The apartment block is on Moscow`s Pyatnitskoye Avenue, in an area dominated by high-rise housing.
The Russian chess team`s national coach, Sergei Yanovsky, said Yeliseyev "was a very talented chess player, a very bright lad, he was always very popular in the team".
"Yura always sought unusual methods in everything, he had a predilection for unorthodox solutions... This is a very heavy loss."
Mr Yanovsky said Yeliseyev "even as a young boy always wanted to show his daring and climb to places.
"But he didn`t go to extremes - he kept within sensible bounds. For example, he`d climb to a height of two metres (6.6ft) and walk along the edge just to show that he had a head for heights."
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