We report on the light sterile neutrino search from the first four-week science run of the KATRIN experiment in 2019. Beta-decay electrons from a high-purity gaseous molecular tritium source are analyzed by a high-resolution MAC-E filter down to 40 eV below the endpoint at 18.57 keV. We consider the framework with three active neutrinos and one sterile neutrino. The analysis is sensitive to the mass, m_{4}, of the fourth mass state for m_{4}^{2}≲1000  eV^{2} and to active-to-sterile neutrino mixing down to |U_{e4}|^{2}≳2×10^{-2}. No significant spectral distortion is observed and exclusion bounds on the sterile mass and mixing are reported. These new limits supersede the Mainz results for m_{4}^{2}≲1000  eV^{2} and improve the Troitsk bound for m_{4}^{2}<30  eV^{2}. The reactor and gallium anomalies are constrained for 100<Δm_{41}^{2}<1000  eV^{2}.