We present a new approach to solving fuel-efficient powered descent guidance problems using the recently developed Theory of Functional Connections. The algorithm is designed to solve the non-linear Two-Point Boundary Value Problem arising from the application of the Pontryagin minimum principle via Chebyshev polynomials expansion of the boundary conditions-free and iterative least-squares method. The proposed algorithm follows under the category of indirect methods for optimal control problems, and it is demonstrated to be fast and accurate, thus potentially suitable for on-board implementation to generate optimal trajectories in real-time. The focus of this paper is on the solution of the equations of motion for the fuel-efficient powered descent guidance via Theory of Functional Connections. We have succeeded in getting solutions at machine error accuracy with just a few iterations, but still suboptimal as the transversality condition for the free-time problem is not yet met.