Topography‐dependent eikonal equation (TDEE) formulated in a curvilinear coordinate system has been recently established and is effective for calculating first‐arrival travel times in an Earth model with an irregular surface. In previous work, the Lax–Friedrichs sweeping scheme used to approximate the TDEE viscosity solutions was only first‐order accurate. We present a high‐order fast‐sweeping scheme to solve the TDEE with the aim of achieving high‐order accuracy in the travel‐time calculation. The scheme takes advantage of high‐order weighted essentially nonoscillatory (WENO) derivative approximations, monotone numerical Hamiltonians, and Gauss Seidel iterations with alternating‐direction sweepings. It incorporates high‐order approximations of the derivatives into the numerical representation of the Hamiltonian such that the resulting numerical scheme is formally high‐order accurate and inherits fast convergence from the alternating sweeping strategy. Extensive numerical examples are presented to verify its efficiency, convergence, and high‐order accuracy.