In this paper, an adaptive neural network-based control method is proposed, using which the problems of uncertainty in marine crane parameters (e.g., payload mass, cart mass, etc.) and uncertain gravity compensation that may lead to positioning errors as well as wave-induced transverse ship rocking motions, which can produce residual oscillations in the payload, are solved. The method does not require any linearization operation, and can effectively control the driven and underdriven state variables so that the cart and the halyard can reach the desired position in a finite time and suppress the residual oscillation of the payload. The stability of the marine crane system at each state equilibrium point is theoretically demonstrated by a rigorous Lyapunov stability analysis. Finally, the practicality and robustness of the control method are verified by simulation.