Spatial extension has been hailed as a ``smoking gun'' in the gamma-ray search of dark galactic subhalos, which would appear as unidentified sources for gamma-ray telescopes. In this work, we study the actual sensitivity of the Fermi-LAT to extended subhalos using simulated data based on a realistic sky model for the first time. We generate spatial templates for a set of representative subhalos, whose parameters were derived from our previous work with N-body cosmological simulation data. We find that detecting an extended subhalo and finding an unequivocal signal of angular extension requires, respectively, a flux 2 to 10 times larger than in the case of a point-like source. By studying a large grid of models, where parameters such as the WIMP mass, annihilation channel or subhalo model are varied significantly, we obtain the response of the LAT as a function of the product of annihilation cross section times the J-factor. Indeed, we show that spatial extension can be used as an additional “filter” to reject subhalos candidates among the pool of unidentified LAT sources, as well as a ``smoking gun'' for positive identification. We conclude that typical angular extensions of about $0.3^\circ$ are expected for the considered scenarios. Finally, we also study the impact of the obtained LAT sensitivity to such extended subhalos on the achievable dark matter constraints, that now worsen by a factor of a few with respect to the point-source limits.