Generative learning is an important task in classical machine learning with several models including generative adversarial networks (GANs) and variational autoencoders (VAEs) which are popular. In quantum machine learning an important task in this setting is that of modeling the distributions obtained by measuring quantum mechanical systems. Classical generative algorithms, including GANs and VAEs, can model the distributions of product states with high fidelity, but fail or require an exponential number of parameters to model entangled states. In this paper, we introduce a quantum-enhanced VAE (QeVAE), a generative quantum-classical hybrid model that uses quantum correlations to improve the fidelity over classical VAEs, while requiring only a linear number of parameters. We provide a closed form expression for the output distributions of the QeVAE. We also empirically show that the QeVAE outperforms classical models on several classes of quantum states, such as 4-qubit and 8-qubit quantum circuit states, haar random states, and quantum kicked rotor states, with a more than 2x increase in fidelity for some states. Finally, we find that the trained model outperforms the classical model when executed on the IBMq Manila quantum computer. As an application we show that our techniques can be also used for the task of circuit compilation. Our work paves the way for new applications of quantum generative learning algorithms and characterizing measurement distributions of high-dimensional quantum states.