The Christianisation of the Roman Empire occurred despite persecution of Christian individuals, suggesting persecution is ineffective in halting conversion to Christianity. However in Japan and China Christianity was persecuted - as far as I can tell in virtually the same ways as the Romans persecuted Christianity - and in these examples it worked very well. Were there differences in the persecution/context I am not aware of?
In Tokugawa Japan, the country was entirely closed to foreigners--with the exception of the Dutch in the very south of the country. The Dutch were not concerned with missionary work, and only wanted access to Japan for trading purposes. In Candide, Voltaire pokes fun at the Dutch, who were forced to walk on crosses in order to access the country. The Japanese saw this as a sign the Dutch weren't devout, but the truth was that the Dutch simply did not place as great importance to symbols.
China, however, has historically had a reasonably strong Christian tradition. One of the first missionaries to enter the country was the Jesuit Matteo Ricci, who developed close relationships with the Chinese. While Christianity never took an incredibly strong hold in China, it did have large impacts on the country overall.
In the wake of the Opium War, there was a man named Hong Xiuquan who was studying hard for the Confucian exams, which had an incredibly low pass rate. After failing the exam twice, he had a nervous breakdown and had vivid dreams. In one dream, he was told that he had to rid China of the "demons." At some point in his life, he had overheard Christian missionaries spreading the Christian faith. He merged Christianity and this dream in his mind, and believed that the voice came from the Christian god. He made the assumption that he was God's son, and therefore the brother of Jesus, and he equated the "demons" to be the ailing Qing Dynasty.
Hong eventually gathered a large group of people and declared war on the Qing, leading to one of the bloodiest civil wars in human history. After some fighting, Hong and his allies (collectively called the Taiping), established a state and conquered Nanjing in the south. The Qing did win the war, although their state was beyond repair.
Furthermore, the Nationalist Party (Guomindang) had a large number of Christians in their ranks. Chiang Kai-shek, for example, was a Christian. This being said, Christianity was not strongly persecuted in China until the emergence of the Communist Party, which suppressed all philosophical-religious movements. Today, China's christian movements are regaining strength, although they are carefully monitored by the CCP.