Large-scale astronomical surveys have the potential to capture data on large numbers of strongly gravitationally lensed supernovae (LSNe). To facilitate timely analysis and spectroscopic follow-up before the supernova fades, an LSN needs to be identified soon after it begins. To quickly identify LSNe in optical survey datasets, we designed ZipperNet, a multi-branch deep neural network that combines convolutional layers (traditionally used for images) with long short-term memory (LSTM) layers (traditionally used for time series). We tested ZipperNet on the task of classifying objects from four categories -- no lens, galaxy-galaxy lens, lensed type Ia supernova, lensed core-collapse supernova -- within high-fidelity simulations of three cosmic survey data sets -- the Dark Energy Survey (DES), Rubin Observatory's Legacy Survey of Space and Time (LSST), and a Dark Energy Spectroscopic Instrument (DESI) imaging survey. Among our results, we find that for the LSST-like dataset, ZipperNet classifies LSNe with a receiver operating characteristic area under the curve of 0.97, predicts the spectroscopic type of the lensed supernovae with 79% accuracy, and demonstrates similarly high performance for LSNe 1-2 epochs after first detection. We anticipate that a model like ZipperNet, which simultaneously incorporates spatial and temporal information, can play a significant role in the rapid identification of lensed transient systems in cosmic survey experiments.