The exact shape of intracranial aneurysms is critical in medical diagnosis and surgical planning. While voxel-based deep learning frameworks have been proposed for this segmentation task, their performance remains limited. In this study, we offer a two-step surface-based deep learning pipeline that achieves significantly better results. Our proposed model takes a surface model of an entire set of principal brain arteries containing aneurysms as input and returns aneurysm surfaces as output. A user first generates a surface model by manually specifying multiple thresholds for time-of-flight magnetic resonance angiography images. The system then samples small surface fragments from the entire set of brain arteries and classifies the surface fragments according to whether aneurysms are present using a point-based deep learning network (PointNet++). Finally, the system applies surface segmentation (SO-Net) to surface fragments containing aneurysms. We conduct a direct comparison of the segmentation performance of our proposed surface-based framework and an existing voxel-based method by counting voxels: our framework achieves a much higher Dice similarity ( $72\%$) than the prior approach ( $46\%$).