Surrogate-based black-box attacks have exposed the heightened vulnerability of DNNs. These attacks are designed to craft adversarial examples for any samples with black-box target feedback for only a given set of samples. State-of-the-art surrogate-based attacks involve training a discriminative surrogate that mimics the target's outputs. The goal is to learn the decision boundaries of the target. The surrogate is then attacked by white-box attacks to craft adversarial examples similar to the original samples but belong to other classes. With limited samples, the discriminative surrogate fails to accurately learn the target's decision boundaries, and these surrogate-based attacks suffer from low success rates. Different from the discriminative approach, we propose a generative surrogate that learns the distribution of samples residing on or close to the target's decision boundaries. The distribution learned by the generative surrogate can be used to craft adversarial examples that have imperceptible differences from the original samples but belong to other classes. The proposed generative approach results in attacks with remarkably high attack success rates on various targets and datasets.