Global routing plays a critical role in modern chip design. The routing paths generated by global routers often form a rectilinear Steiner tree (RST). Recent advances from the machine learning community have shown the power of learning-based route generation; however, the yielded routing paths by the existing approaches often suffer from considerable overflow, thus greatly hindering their application in practice.We propose NeuralSteiner, an accurate approach to overflow-avoiding global routing in chip design. The key idea of NeuralSteiner approach is to learn Steiner trees: we first predict the locations of highly likely Steiner points by adopting a neural network considering full-net spatial and overflow information, then select appropriate points by running a graph-based post-processing algorithm, and finally connect these points with the input pins to yield overflow-avoiding RSTs. NeuralSteiner offers two advantages over previous learning-based models. First, by using the learning scheme, NeuralSteiner ensures the connectivity of generated routes while significantly reducing congestion. Second, NeuralSteiner can effectively scale to large nets and transfer to unseen chip designs without any modifications or fine-tuning. Extensive experiments over public large-scale benchmarks reveal that, compared with the state-of-the-art deep generative methods, NeuralSteiner achieves up to a 99.8\% reduction in overflow while speeding up the generation and maintaining a slight wirelength loss within only 1.8\%.