We report the discovery of two ultra-faint stellar systems found in early data from the DECam Local Volume Exploration survey (DELVE). The first system, Centaurus I (DELVE J1238─4054), is identified as a resolved overdensity of old and metal-poor stars with a heliocentric distance of ${\text{}}{D}_{\odot }={116.3}_{-0.6}^{+0.6}\,\mathrm{kpc}$ , a half-light radius of ${r}_{h}={2.3}_{-0.3}^{+0.4}\,\mathrm{arcmin}$ , an age of $\tau \gt 12.85\,\mathrm{Gyr}$ , a metallicity of $Z={0.0002}_{-0.0002}^{+0.0001}$ , and an absolute magnitude of ${M}_{V}=-{5.55}_{-0.11}^{+0.11}\,\mathrm{mag}$ . This characterization is consistent with the population of ultra-faint satellites and confirmation of this system would make Centaurus I one of the brightest recently discovered ultra-faint dwarf galaxies. Centaurus I is detected in Gaia DR2 with a clear and distinct proper motion signal, confirming that it is a real association of stars distinct from the Milky Way foreground; this is further supported by the clustering of blue horizontal branch stars near the centroid of the system. The second system, DELVE 1 (DELVE J1630─0058), is identified as a resolved overdensity of stars with a heliocentric distance of ${\text{}}{D}_{\odot }={19.0}_{-0.6}^{+0.5}\,\mathrm{kpc}$ , a half-light radius of ${r}_{h}={0.97}_{-0.17}^{+0.24}\,\mathrm{arcmin}$ , an age of $\tau ={12.5}_{-0.7}^{+1.0}\,\mathrm{Gyr}$ , a metallicity of $Z={0.0005}_{-0.0001}^{+0.0002}$ , and an absolute magnitude of ${M}_{V}=-{0.2}_{-0.6}^{+0.8}\,\mathrm{mag}$ , consistent with the known population of faint halo star clusters. Given the low number of probable member stars at magnitudes accessible with Gaia DR2, a proper motion signal for DELVE 1 is only marginally detected. We compare the spatial position and proper motion of both Centaurus I and DELVE 1 with simulations of the accreted satellite population of the Large Magellanic Cloud (LMC) and find that neither is likely to be associated with the LMC.