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 D⊙ =116.3-0.6+0.6, a half-light radius of rh = 2.3-0.3 +0.4 arcmin, an age of τ > 12.85 Gyr, a metallicity of Z = 0.0002-0.0002 +0.0001, and an absolute magnitude of MV = -5.55-0.11 +0.11. 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 D⊙ = 19.0-0.6 +0.5 kpc, a half-light radius of rh = 0.97-0.17 +0.24 arcmin, an age of τ = 12.5-0.7 +1.0 Gyr, a metallicity of Z = 0.0005-0.0001 +0.0002, and an absolute magnitude of MV = 0.2-0.6 +0.8, 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.