We present a direct search for cosmic relic neutrinos using data acquired during the first two science runs of the KATRIN experiment in 2019. Beta decay electrons originating from a high-purity molecular tritium gas source are analyzed by a high-resolution MAC-E filter near the endpoint at \SI{18.57}{keV}. This measurement is sensitive to a local relic neutrino overdensity ratio of $ \eta < $ \num{9.7e10}/$\alpha$ (\num{1.1e11}/$\alpha$) at a confidence level of 90\% (95\%) with $\alpha=1$ (0.5) for Majorana neutrinos (Dirac). A adjustment of the integrated electron tritium spectrum over a restricted area around the endpoint, taking into account the relic neutrino captures in the tritium source, yields no significant overdensity. This achievement is an improvement of the previous neutrino mass experiments at Los Alamos and Troitsk.