The purpose of this paper is to compute the minimal fibering degree of an arbitrary projective toric variety. We prove that it equals the lattice width of the associated polytope. This gives a complete answer to a question asked in a recent paper of Levinson, Ullery and the second author. The minimal fibering degree of a polarized projective variety was introduced in that paper in order to compute the degree of irrationality (a generalization of gonality) of high degree divisors. From this perspective, our paper gives a higher dimensional analogue of results of Kawaguchi and others who computed the gonality of curves in toric surfaces in terms of lattice widths.