In this paper, a new zeroing neural network (NZNN) with a new activation function (AF) is presented and investigated for solving dynamic Sylvester equation (DSE). The proposed NZNN not only finds the solutions of the DSE in fixed-time but also has better robustness, and its superior effectiveness and robustness are proved by rigorous mathematical analysis. Numerical simulation results of the proposed NZNN, the original zeroing neural network activated by other recently reported AFs and the existing robust nonlinear ZNN (RNZNN) for solving second-order dynamic Sylvester equation and third-order dynamic Sylvester equation are provided for the purpose of comparison. Comparing with the existing ZNN models, the proposed NZNN has better robustness and faster convergence performance for solving DSE in the same noise environment. Moreover, a successful robot manipulator trajectory tracking example in noise-disturbed environment using the proposed NZNN is also applied for illustrating its further practical applications.