How did the Tang & Song Dynasties maintain and justify their rule?

Through the principle of Mandate of Heaven.

Explanation:

Mandate of Heaven refers to the belief that whoever become the rulers in China obtain the position because the Gods intended them to. So, they believe that if one ruler is killed and changed by another rulers, it happened because the initial ruler is deemed as 'unworthy' by the heaven to continue his reign.

Many people in Tang & song dynasties justify the cruel things that they did through this belief. The see that the word of the emperor is the word from the heaven that must be obeyed by all citizens.