Despite decades of research, identifying selective sweeps, the genomic footprints of positive selection, remains a core problem in population genetics. Of the myriad methods that have been made towards this goal, few are designed to leverage the potential of genomic time-series data. This is because in most population genetic studies only a single, brief period of time can be sampled for a study. Recent advancements in sequencing technology, including improvements in extracting and sequencing ancient DNA, have made repeated samplings of a population possible, allowing for more direct analysis of recent evolutionary dynamics. With these advances in mind, here we present Timesweeper, a fast and accurate convolutional neural network-based tool for identifying selective sweeps in data consisting of multiple genomic samplings of a population over time. Timesweeper utilizes this serialized sampling of a population by first simulating data under a demographic model appropriate for the data of interest, training a 1D Convolutional Neural Network on said model, and inferring which polymorphisms in this serialized dataset were the direct target of a completed or ongoing selective sweep. We show that Timesweeper is accurate under multiple simulated demographic and sampling scenarios, and identifies selected variants with impressive resolution. In sum, we show that more accurate inferences about natural selection are possible when genomic time-series data are available; such data will continue to proliferate in coming years due to both the sequencing of ancient samples and repeated samplings of extant populations with faster generation times. Methodological advances such as Timesweeper thus have the potential to help resolve the controversy over the role of positive selection in the genome. We provide Timesweeper both as a Python package and Snakemake workflow for use by the community.