Inspired by the unidirectional convergence of water in a funnel, we propose a vertical diode architecture that integrates field-effect gating, rectification, and memory functionalities. This design enables facile resistive switching of functional molecular monolayers, bidirectional electric field modulation, and low-voltage nondestructive readout. The core of the funnel diode consists of a 3.6 nm thick nanopore-decorated dielectric layer and a 10-30 nm thick unipolar organic semiconductor (e.g., pentacene), sandwiched between two parallel-plate Ohmic electrodes. When majority carriers are injected from the semiconductor side, the current exceeds that from the opposite direction by a rectifying ratio exceeding 104, due to the gating effect. By inserting molecular monolayers of Cu(II) stearate or Au25-clusters between the dielectric and pentacene layers, the funnel diode demonstrates both nonvolatile memory performance and synaptic-mimic behavior under ultralow read voltages (e.g., ±0.05 V). This funnel diode architecture fully inherits the advantages of field-effect transistors while circumventing their intrinsic limitations. Moreover, it offers a promising strategy to translate molecular or atomic-scale state changes in two-dimensional materials into resistance changes, paving the way for next-generation hardware architectures in post-Moore era.