The discrete exterior calculus (DEC) is a theory of exterior calculus defined for discrete manifolds. The DEC has been used to discretize physical systems so that various geometric and physical properties are preserved from the continuous PDE system. Here we detail a formulation of the DEC for cubical complexes of two and three dimensions and then demonstrate an implementation of compressible Navier-Stokes (NS) in a low-Mach regime within this formulation. There has been previous work in defining DEC operators on two-dimensional Cartesian grids of constant height and width. Separately, there has been work on developing select DEC operators for various primal-dual three-dimensional lattices specifically for the Gross-Pitaevskii equation and Maxwell equations. However, these lack definitions of the Lie derivative which are critical for modeling the hyperbolic terms in NS. The closest to our work is an implementation of two-phase incompressible NS within the framework of the DEC. We base our implementation of NS in the DEC on previous work demonstrating an exterior calculus formulation of the Euler equations. We begin with an overview of the DEC for cubical complexes by detailing operator definitions for two and three dimensions. We then introduce a formulation of the low-Mach compressible NS equations within the exterior calculus and afterwards present its spatial and temporal discretization. Finally, we demonstrate the validity of the discretization through various physical examples including the rising thermal bubble and Rayleigh-Taylor instabilities. Code for high-performance implementations of the cubical DEC is included in CombinatorialSpaces.jl, which is an open-source Julia library that hitherto implemented the DEC solely over simplicial complexes.