Why did Deng Xiaoping survive the 'crisis of communism' whilst Mikhail Gorbachev did not?
The most likely reason for this is that Deng did not try to overhaul China's system as radically as Gorbachev tried to overhaul that of the Soviet Union.
In the Soviet Union, Khrushchev tried a complete set of reforms. He tried to reform the economy through perestroika, but he also tried to open the political system through glasnost. By contrast, Deng only tried to reform the Chinese economy. Deng did say that "to get rich is glorious" and he did push for more capitalistic policies. However, he did not do anything to open up the Chinese political system. He supported political repression, including the use of violence in Tiananmen Square in 1989.
These sort of repressive action ensured that there would not be too many dissidents pushing to break China's system. At the same time, the success of Deng's economic policies also helped to keep people happy. In these ways, Deng was able to stay in power through the "crisis of communism."