The 25-year-old pianist picked up a Â£20,000 cheque at Leeds Town Hall.
The competition takes place every three years and is open to pianists under the age of 30 from around the world.
This was the last contest to be overseen by Dame Fanny Waterman, who co-founded the event in 1961 and has stepped down at the age of 95.
Dame Fanny said: "This has been a truly marvellous competition with some of the finest young pianists entering.
"To hear a talent as fine as Anna's brings confidence that the competition will continue to produce the greatest pianists in the world for many years to come."
This year's six finalists performed with the Halle Orchestra, led by its music director Sir Mark Elder.
Sir Mark said: "Once again the Leeds International Piano Competition has found a truly exciting winner from some very talented finalists.
"The competition continues to achieve the highest standards and I am delighted to play a small part in it."
Tcybuleva is only the second female winner, and triumphed with Brahms' Concerto No 2 in B flat major. Her prize also included the Daw Aung San Suu Kyi gold medal and the Champs Hill Records award, allowing her to record her debut solo CD.
However her performance did not impress all observers.
Writing on the Slipped Disc blog, Erica Worth, editor of Pianist magazine, said: "This work needs warmth, gravitas, a certain humility, a feeling that one has lived, not to mention a velvety rich tone and utter command over the keyboardâ€¦
"But sadly, for this writer, it didn't deliver. There were some memory lapses and wrong notes too."
The Guardian critic Andrew Clements wrote that Tcybuleva's success "certainly raised a few eyebrows".
He said: "For all the fluency of her playing, she often seemed incapable of seeing the overall shape of the work, and her role in projecting it, rather than the detail of each passing moment."
Second place went to Heejae Kim, 28, from South Korea, with third prize going to Vitaly Pisarenko, 28, from Russia.
The contest was broadcast on BBC Radio 3 on Sunday, and will be screened on BBC Four on the next three consecutive Friday evenings.
Russian Anna Tcybuleva has won the Leeds International Piano Competition, one of the most prestigious prizes in classical music.