The issue of monitoring and early warning of rock instability has received increasing critical attention in the study of rock engineering. To investigate the damage evolution process of granite under triaxial compression tests, acoustic emission (AE) tests were performed simultaneously. This study firstly introduced two novel parameters, i.e., the coefficient of variation (CoV) of the information entropy and correlation dimension of the amplitude data from the AE tests, to identify the precursor of the failure of granite. Then the relationship between the changes in these parameters and the stress-time curve was compared and analyzed. The results of this study show that: (1) There is a strong correlation between the CoV of the information entropy and the failure process of granite. The granite failed when the CoV curve raised to a plateau, which could be used as an indicator of rock instability. (2) The fluctuation of the correlation dimension indicates the different stages during the loading process, i.e., the initial compaction stage, the linear elastic stage, the yield stage, and the failure stage. Each stage contains a descending and a rising process in the correlation dimension curve, and the exhibited starting point or the bottom point at the correlation dimension curve could be selected as the indicator point for the rock instability. (3) The combined analysis of the Information entropy and Correlation dimension can improve the accuracy of rock instability prediction. This study provides new insights into the prediction of rock instability, which has theoretical implications for the stability of subsurface engineering rock masses.