We present the first Open Gravitational-wave Catalog, obtained by using the public data from Advanced LIGO's first observing run to search for compact-object binary mergers. Our analysis is based on new methods that improve the separation between signals and noise in matched-filter searches for gravitational waves from the merger of compact objects. The three most significant signals in our catalog correspond to the binary black hole mergers GW150914, GW151226, and LVT151012. We assume a common population of binary black holes for these three signals by defining a region of parameter space that is consistent with these events. Under this assumption, we find that LVT151012 has a 97.6% probability of being astrophysical in origin. No other significant binary black hole candidates are found, nor did we observe any significant binary neutron star or neutron star-black hole candidates. We make available our complete catalog of events, including the subthreshold population of candidates.