Resistive crossbars implement parallel vector-matrix multiplication (VMM) in analog fashion, and thus enable fast and energy-efficient neuromorphic systems. However, interconnect resistance and resistive switching devices form a complex resistance network with sneak paths. It could result in severe distortions on the output currents. When implementing neural networks, current distortions also cause significant accuracy loss. This article proposes an accurate and computationally efficient model of VMM in resistive crossbars, called diagonal matrix regression (DMR), and incorporates the model into the topology of neural networks as DMR layer (DMRL). Given an <inline-formula> <tex-math notation="LaTeX">{m}{\times }{n} </tex-math></inline-formula> crossbar, two diagonal matrices are calculated directly according to the resistance network in a time complexity of only <inline-formula> <tex-math notation="LaTeX">{O}({m}^{2}+{n}^{2}) </tex-math></inline-formula>. No hyper-parameter needs to be determined manually. Modeling of VMM is implemented in a time complexity of only <inline-formula> <tex-math notation="LaTeX">{O}({mn}) </tex-math></inline-formula>. DMRL is developed to replace the weight matrix of neural networks so that the effect of interconnect resistance and the sneak path problem are well handled during ex-situ training. Using this technique, for the task of MNIST and fashion-MNIST classification, the accuracy is dramatically restored.