Future long-baseline experiments will be able to probe hitherto unexplored regions of sterile neutrino parameter space for masses ranging from meV to eV. We present an analytic calculation of the neutrino conversion probability $P(\nu_\mu \to \nu_e)$ in the presence of sterile neutrinos, with exact dependence on $\Delta m^2_{41}$ and matter effects. We further express the neutrino conversion probability as a sum of terms of the form $\sin(x)/x$, thus allowing a physical understanding of matter effects and their possible resonance-like behavior. We focus on the identification of sterile mass ordering (sign of $\Delta m^2_{41}$) at DUNE. The conversion probability obtained reveals the complex interplay between sterile and matter contributions. We perform numerical calculations of DUNE's sensitivity to sterile mass ordering over a broad range of sterile neutrino masses. Our analytic expressions enable us to explain the dependence of this sensitivity on $\Delta m^2_{41}$ values for all mass ordering combinations.