The disturbance rejection control problem is studied in this article for the 2-D Roesser system within 2-D event-triggered control frameworks. First, the unknown disturbance is constructed as an extended state, such that both the system states and the unknown disturbance can be estimated by the given 2-D extended state observer (ESO). Through the 2-D event-triggered mechanisms, ESO-based feedback controllers are designed. By Lyapunov theory, a common analysis method and a directional analysis method are given to obtain the sufficient conditions for the system stability, respectively. Then, genetic algorithm is applied to avoid introducing conservatism in the solving process of the given conditions. It not only prominently reduces the utilization of communication resources, but also realizes a boundary-based tradeoff between the horizontal state and the vertical state, as well as achieves the disturbance compensation. Finally, numerical simulations are given to demonstrate the validity of the proposed control framework.