In this work, we present classification results on early supernova lightcurves from SCONE, a photometric classifier that uses convolutional neural networks to categorize supernovae (SNe) by type using lightcurve data. SCONE is able to identify SN types from lightcurves at any stage, from the night of initial alert to the end of their lifetimes. Simulated LSST SNe lightcurves were truncated at 0, 5, 15, 25, and 50 days after the trigger date and used to train Gaussian processes in wavelength and time space to produce wavelength-time heatmaps. SCONE uses these heatmaps to perform 6-way classification between SN types Ia, II, Ibc, Ia-91bg, Iax, and SLSN-I. SCONE is able to perform classification with or without redshift, but we show that incorporating redshift information improves performance at each epoch. SCONE achieved 75% overall accuracy at the date of trigger (60% without redshift), and 89% accuracy 50 days after trigger (82% without redshift). SCONE was also tested on bright subsets of SNe (r < 20 mag) and produced 91% accuracy at the date of trigger (83% without redshift) and 95% 5 days after trigger (94.7% without redshift). SCONE is the first application of convolutional neural networks to the early-time photometric transient classification problem. All of the data processing and model code developed for this paper can be found in the SCONE software package located at github.com/helenqu/scone (Qu 2021).