In this paper, 3-D vector field tomography (3D-VFT) is employed to reconstruct three-dimensional, irrotational fields in a bounded cubic domain. A sampling process along the scanning lines that further assigns the derived points to preordained finite reconstruction points accomplishes data redundancy, lacking when the problem is formed in the continuous domain, and results in the formulation of an over-determined system of linear equations. The only precondition to the system solution, that corresponds to a discretized inversion of the Ray transform, is the known location and values of a limited number of boundary points. The method is accompanied by a theoretical analysis on the regularization achieved and the errors introduced. The effectiveness and robustness of the method are demonstrated by means of simulations of electric fields, a series of perturbation tests, and a comparison with two alternative baseline methodologies.