In this paper, we solve the nonlinear \(H_\infty\) optimal control with output feedback via the neural network (NN)–least squares method for the affine nonlinear system. The approach is based on successive approximate solution of two Hamilton–Jacobi–Isaacs (HJI) equations, which appear in the \(H_\infty\) optimal output feedback control. Successive approximation (SA) approach combined with neural network (NN) for updating control and disturbance inputs in the case of state-feedback control is first proposed to solve an HJI equation with two players. The obtained solution is then used to solve, with the SA-NN approach for updating disturbance input, an HJI equation with one player in the output feedback control problem. Simulations on the Translational oscillator with rotational actuator mechanical system are presented to illustrate the effectiveness of the proposed method.