We present RAPID (Real-time Automated Photometric IDentification), a novel time-series classification tool capable of automatically identifying transients from within a day of the initial alert, to the full lifetime of a light curve. Using a deep recurrent neural network with Gated Recurrent Units (GRUs), we present the first method specifically designed to provide early classifications of astronomical time-series data, typing 12 different transient classes. Our classifier can process light curves with any phase coverage, and it does not rely on deriving computationally expensive features from the data, making RAPID well-suited for processing the millions of alerts that ongoing and upcoming wide-field surveys such as the Zwicky Transient Facility (ZTF), and the Large Synoptic Survey Telescope (LSST) will produce. The classification accuracy improves over the lifetime of the transient as more photometric data becomes available, and across the 12 transient classes, we obtain an average area under the receiver operating characteristic curve of 0.95 and 0.98 at early and late epochs, respectively. We demonstrate RAPID's ability to effectively provide early classifications of transients from the ZTF data stream. We have made RAPID available as an open-source software package (this https URL) for machine learning-based alert-brokers to use for the autonomous and quick classification of several thousand light curves within a few seconds.