Two-time Olympic medallist Page won the women's competition, superbly executing a routine high on difficulty to score 56.235.
Neither of the two remaining athletes, Chinese pair Huang Yanfei and Cao Yunzhu, could better her score with Cao's 55.815 earning her a second silver medal of the weekend after China's second in the team event.
Iana Lebedeva (55.800) was third after helping RGF to team bronze.
After winning world team gold in 2013, this is 30-year-old Page's first global individual title after overcoming 'lost move syndrome' - similar to the yips or twisties - early in her career.
The men's final was missing reigning Olympic champion Ivan Litvinovich who was unable to complete his routine in Saturday's semi-finals.
Yan was top in qualifying, and repeated that result in the final thanks to some supremely impressive execution.
The 22-year-old's score of 61.825 saw him clinch a comfortable victory from Japan's Nishioka Ryusei who had a difficulty tariff 1.100 points higher than Yan but lost marks for execution.
Aleh Rabtsau (60.385) took bronze for Belarus with New Zealand's Olympic bronze medallist Dylan Schmidt only seventh.
Rabtsau had already won team gold for Belarus, with Litvinovich, and synchro gold with Andrei Builou.
Aleksandr Lisitsyn retained his individual tumbling title having led RGF to team gold.
Britain's Megan Kealy took the women's crown to go with her team bronze, with Lucie Tumoine settling for silver after winning team gold with France.