Noise is ubiquitous in the various real and artificial systems, which plays multiple effects on synchronization of nonlinear systems. This study proposes to explore the positive influence of external common noise on the synchronous behavior of high dimensional discrete complex network. A network model of two uncoupled discrete chaotic networks with identical parameters under the excitation of external common noise is firstly constructed. By using the Birkhoff traversal theorem and the matrix theory, some sufficient conditions for synchronization of two discrete chaotic network systems are then proved. Meanwhile, by employing a specific chaotic network model, numerical simulation is used to verify the effectiveness of the theoretical analysis. The numerical results illustrate that when the parameters of network model meet the sufficient conditions for common noise induced synchronization, the two discrete chaotic networks can achieve synchronization in random sense. Moreover, the synchronization is robust against the variation of complex network topology. This paper obtains the theoretical conditions for common noise induced synchronization in the discrete chaotic networks for the first time, which not only can enrich the research of noise induced synchronization in some extent, but also can help to understand the positive effect of noise on ordered dynamics in highdimensional nonlinear chaotic systems.