The evolution of human-specific lateralised functions such as language has been linked to the development of structural asymmetries in the brain. Here we applied state of the art image analysis techniques to measure Sylvian Fissure (SF) asymmetry and Occipital Bending (OB) in 3D Magnetic Resonance (MR) images of the brain obtained in-vivo for 27 humans and 29 chimpanzees (Pan troglodytes). SF morphology differed between species, with the human SF terminating more superiorly in right inferior parietal lobe, an asymmetry that was on average absent in chimpanzees (F (1,52) = 10.692, p = 0.002). Irrespective of morphology, Total SF Length was, as previously reported, leftward in humans but not in chimpanzees, although the difference did not reach significance between species. However, when only brains possessing comparable bilateral SF bifurcation morphology were compared, humans showed previously reported left-lateralised Anterior-Horizontal (AH-SF) and right-lateralised Vertical (V-SF) SF asymmetries. In contrast, chimpanzees lacked both asymmetries, and this approached significant difference between-species in the AH-SF segment (F (1, 34) = 2.365, p = 0.059). On average in humans the left occipital lobe crossed the midline toward the right (Rightward OB) which was significantly different from the chimpanzee cohort that showed no average OB (Independent-Samples Mann-Whitney U Test, p = 0.012). Furthermore, OB was related to SF asymmetry in humans, such that the more rightward V-SF and leftward AH-SF, the more rightward the OB. This pattern of SF and OB asymmetries was found in 41.7% of human individuals with bilateral SF bifurcation but none of the chimpanzees. To our knowledge, this is the first study highlighting that a pattern of SF and OB asymmetry distinguishes the human from the chimpanzee brain, and suggests this may be associated with a unique trajectory of brain development and functional abilities in humans.