We present a F160W morphological catalog for the COSMOS-DASH survey, the largest area near-IR survey using HST-WFC3 to date. Utilizing the “Drift And SHift” (DASH) observing technique for NIR imaging, in which guide star acquisition is only done once per orbit and instrument drift is corrected in post processing, the COSMOS-DASH survey imaged approximately 0.7 square degrees of the UltraVISTA deep stripes. Global structural parameters are measured for about 50,000 galaxies within COSMOS-DASH using GALFIT, with detection using a deep combined optical and near-infrared noise-equalized HST image. While the shallower depth of DASH observations and the non-standard data reduction techniques may not be ideal for low-mass or extended galaxies, we recover consistent results with those from the deeper 3D-HST morphological catalogs out to z=2 and log(M/Msun)~10. As the DASH technique limits each pointing to an exposure time of ~300 seconds, the corresponding images are shallower than usual HST NIR imaging with 5-sigma depths of 25.1 ABmag. Here we test the parameter space where morphological fits of DASH galaxies are robustly recovered. In general, sizes and Sérsic indices of typical galaxies are accurately obtained for H(160)<23 and H(160)<22. In size-mass parameter space, galaxies in COSMOS-DASH demonstrate robust morphologies in general below redshift 2 and log(M/Msun)>9. At log(M/Msun)<10.5, we find a flattening of the quiescent size-mass relation, implying the sizes of low mass quiescent galaxies are more similar to star-forming galaxies with a similar redshift and mass. Trends with axis-ratio and environment suggest that low mass galaxies that quench as satellites may retain morphologies comparable to similar mass star-forming galaxies.