Our ability to identify faces is thought to depend on high-level visual processing in the brain. Nonetheless, studies of face recognition have generally relied on 2D face photographs where low-level strategies relying on texture and appearance cues can be employed to adequately support high face identification performance. Here, we designed a fine face discrimination task under 3D pose and lighting variation that was purely based on shape, a task which challenged state-of-the-art artificial vision systems compared to object recognition tasks. In contrast, humans performed this shape based face task at comparable levels to their object recognition performance. We then tested one of the smallest simian primates on this human-level, machine-difficult visual task, the common marmoset -- a small, New World monkey. Marmosets successfully discriminated between face identities across 3D viewing conditions based purely on face shape. Their face recognition performance was on par with their object recognition performance and exhibited face-specific behavioral signatures similar to humans, including lower performance for inverted faces, faces lit from below, and contrast reversed faces. These results demonstrate that a high-level visual behavior, invariant face recognition based purely on geometry and not additional texture and appearance cues, is shared across simian primates from among the smallest to the most advanced, consistent with the presence of common underlying high-level visual brain areas across simian primates.