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Organic synaptic memristors are of considerable interest owing to their attractive characteristics and potential applications to flexible neuromorphic electronics. In this work, an organic type-II heterojunction consisting of poly(3,4-ethylenedioxythiophene): polystyrene sulfonate (PEDOT:PSS) and pentacene was adopted for low-voltage and flexible memristors. The conjugated polymer PEDOT:PSS serves as the flexible resistive switching (RS) layer, while the thin pentacene layer plays the role of barrier adjustment. This heterojunction enabled the memristor device to be triggered with low-energy RS operations (V < ± 1.0 V and I < 9.0 μA), and simultaneously providing high mechanical bending stability (bending radius of ≈2.5 mm, bending times = 1,000). Various synaptic properties have been successfully mimicked. Moreover, the memristors presented good potentiation/depression stability with a low cycle-to-cycle variation (CCV) of less than 8%. The artificial neural network consisting of this flexible memristor exhibited a high accuracy of 89.0% for the learning with MNIST data sets, even after 1,000 tests of 2.5% stress-strain. This study paves the way for developing low-power and flexible synaptic devices utilizing organic heterojunctions.