We use a $G_2$-structure on a 7-dimensional Riemannian manifold to define an octonion bundle with a fiberwise non-associative product. We then define a metric-compatible octonionic covariant derivative on this bundle that is also compatible with the octonion product. The torsion of the $G_2$-structure is then shown to be an octonionic connection for this covariant derivative with curvature given by the component of the Riemann curvature that lies in the 7-dimensional representation of $G_2$. The choice of a particular $G_2$-structure within the same metric class is then interpreted as a choice of gauge and we show that under a change of this gauge, the torsion transforms as an octonion-valued connection 1-form. We then also define an energy functional for octonion sections, the critical points of which are shown to correspond to an octonionic analog of the Coulomb gauge. The gradient flow for this functional is an octonionic harmonic map heat flow that minimizes the torsion within the same metric class.