The calculation of a three-dimensional underwater acoustic field has always been a key problem in computational ocean acoustics. Traditionally, this solution is usually obtained by directly solving the acoustic Helmholtz equation using a finite difference or finite element algorithm. Solving the three-dimensional Helmholtz equation directly is computationally expensive. For quasi-three-dimensional problems, the Helmholtz equation can be processed by the integral transformation approach, which can greatly reduce the computational cost. In this paper, a numerical algorithm for a quasi-three-dimensional sound field that combines an integral transformation technique, stepwise coupled modes and a spectral method is designed. The quasi-three-dimensional problem is transformed into a two-dimensional problem using an integral transformation strategy. A stepwise approximation is then used to discretize the range dependence of the two-dimensional problem; this approximation is essentially a physical discretization that further reduces the range-dependent two-dimensional problem to a one-dimensional problem. Finally, the Chebyshev--Tau spectral method is employed to accurately solve the one-dimensional problem. We provide the corresponding numerical program SPEC3D for the proposed algorithm and describe several representative numerical examples. In the numerical experiments, the consistency between SPEC3D and the analytical solution/high-precision finite difference program COACH verifies the reliability and capability of the proposed algorithm. A comparison of running times illustrates that the algorithm proposed in this paper is significantly faster than the full three-dimensional algorithm in terms of computational speed.