This paper investigates the problem of path following of underactuated marine surface vessels (MSVs) with uncertain nonlinear dynamics. First, the tracking target of the vessel is reduced from tracking the earth-fixed position to tracking the line-of-sight (LOS) angle by LOS method. Then, by employing the radial basis function neural network (RBFNN) to deal with the uncertain nonlinear dynamics, an adaptive NN fast power reaching law is developed for the path following problem based on the backstepping design methodology. Thereafter, the event-triggered technique is incorporated into the control design to synthesize an adaptive NN event-triggered controller with the fast power reaching convergence rate. By combining with the presented event-triggered mechanism, the controller is only updated when the triggering condition is satisfied. Therefore, both the update frequency of the controller and actuator loss are greatly reduced comparing with the traditional time-triggered controller. Theoretical analysis via Lyapunov method indicates that the tracking error can converge to zero within a finite time, meanwhile it also shows that Zeno behavior can be avoided. Simulation results with comparations illustrate the validity and superiority of the proposed controller.