The first observations of B0→D¯∗(2007)0K+π- and Bs0→D¯∗(2007)0K-π+ decays are presented, and their branching fractions relative to that of the B0→D¯∗(2007)0π+π- decay are reported. These modes can potentially be used to investigate the spectroscopy of charm and charm-strange resonances and to determine the angle γ of the Cabibbo-Kobayashi-Maskawa unitarity triangle. It is also important to understand them as a source of potential background in determinations of γ from B+→DK+ and B0→DK+π- decays. The analysis is based on a sample corresponding to an integrated luminosity of 5.4 fb-1 of proton-proton collision data at 13 TeV center-of-mass energy recorded with the LHCb detector. The D¯∗(2007)0 mesons are fully reconstructed in the D¯0π0 and D¯0γ channels with the D¯0→K+π- decay. A novel weighting method is used to subtract background while simultaneously applying an event-by-event efficiency correction to account for resonant structures in the decays.