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Single-cell RNA-sequencing (scRNA-seq) enables high throughput measurement of RNA expression in individual cells. Due to technical limitations, scRNA-seq data often contain zero counts for many transcripts in individual cells. These zero counts, or dropout events, complicate the analysis of scRNA-seq data using standard analysis methods developed for bulk RNA-seq data. Current scRNA-seq analysis methods typically overcome dropout by combining information across cells, leveraging the observation that cells generally occupy a small number of RNA expression states. We introduce netNMF-sc, an algorithm for scRNA-seq analysis that leverages information across both cells and genes. netNMF-sc combines network-regularized non-negative matrix factorization with a procedure for handling zero inflation in transcript count matrices. The matrix factorization results in a low-dimensional representation of the transcript count matrix, which imputes gene abundance for both zero and non-zero entries and can be used to cluster cells. The network regularization leverages prior knowledge of gene-gene interactions, encouraging pairs of genes with known interactions to be close in the low-dimensional representation. We show that netNMF-sc outperforms existing methods on simulated and real scRNA-seq data, with increasing advantage at higher dropout rates (e.g. above 60%). Furthermore, we show that the results from netNMF-sc -- including estimation of gene-gene covariance -- are robust to choice of network, with more representative networks leading to greater performance gains.