Extensive research is directed towards neuromorphic computing, a hardware-based processing paradigm based on the human brain, to satisfy the need for more efficient computing. State-of-the-art resistive switching devices, also called memristors, use 2D transition metal dichalcogenide materials as a switching material and can potentially be used as next-generation neuromorphic devices. Chemical vapor deposition produced two-dimensional transition metal dichalcogenides are the most employed in resistive switching devices since the layers contain defects like vacancies and grain boundaries that can induce resistive switching effects. However, only a few studies have investigated resistive switches based on exfoliated layers and exploited the superior crystalline quality. In this work, a vertical memristor based on exfoliated molybdenum disulfide is demonstrated, which reveals a Schottky-barrier-height-modulation resistive switching mechanism. The devices exhibit a gradual, forming-free resistive switching characteristic. We attribute the switching mechanism to either the mobility of sulfur vacancies caused by a voltage bias applied to the device or charge trapping, which modulates the molybdenum disulfide/metal barrier. We present the characteristic quasi-static I-V behavior, synaptic potentiation and depression, demonstrating the potential of being a basic device to realize synapses for neuromorphic systems.