We present the H_160 morphological catalogs for the COSMOS-DASH survey, the largest area near-IR survey using HST-WFC3 to date. Utilizing the “Drift And SHift” observing technique for HST-WFC3 imaging, the COSMOS-DASH survey imaged approximately 0.5 deg^2 of the UltraVISTA deep stripes (0.7 deg^2, when combined with archival data). Global structural parameters are measured for 51,586 galaxies within COSMOS-DASH using GALFIT (excluding the CANDELS area) with detection using a deep multi-band HST image. We recover consistent results with those from the deeper 3D-HST morphological catalogs, finding that, in general, sizes and Sérsic indices of typical galaxies are accurate to limiting magnitudes of H_160 < 23 and H_160 < 22 ABmag, respectively. In size-mass parameter space, galaxies in COSMOS-DASH demonstrate robust morphological measurements out to z~2 and down to log(M_⋆/M_⊙)∼ 9 . With the advantage of the larger area of COSMOS-DASH, we measure a flattening of the quiescent size-mass relation below log(M_⋆/M_⊙)∼ 10.5 that persists out to z~2. We show that environment is not the primary driver of this flattening, at least out to z = 1.2, whereas internal physical processes may instead govern the structural evolution.