For organic memristors, non-zero-crossing current−voltage (I− V) curves are often observed, which can be attributed to capacitive effects. If the conversion between the capacitance-dominated state (CDS) and the memristance-dominated state (MDS) can be realized in a controllable manner, more device functions can be obtained. In this work, a two-terminal memristor using a common organic dye, azobenzene (AZB), as the active layer was prepared. It is found that as the applied voltage gradually increases, the device can transition from CDS to MDS. In the low voltage range (<1 V), the device is in CDS, and the capacitance is significantly increased by ∼10 4 compared to the theoretical value. In the high voltage range (>1 V), the device is in MDS, achieving an HRS (high resistance state)/LRS (low resistance state) resistance ratio of ∼10 4 , and the logic operations are achieved. Through the analysis of the I−V curve, energy diagram of the materials, and computer simulation results, the mechanisms of CDS, MDS, and their conversion are proposed. This work provides an in-depth understanding of the working mechanism of organic memristors and demonstrates the potential of AZB-based organic memristors for information storage and logic display applications.