We present DeepStreaks, a convolutional-neural-network, deep-learning system designed to efficiently identify streaking fast-moving near-Earth objects that are detected in the data of the Zwicky Transient Facility (ZTF), a wide-field, time-domain survey using a dedicated 47 sq. deg camera attached to the Samuel Oschin 48-inch Telescope at the Palomar Observatory in California, United States. The system demonstrates a 96-98% true positive rate, depending on the night, while keeping the false positive rate below 1%. The sensitivity of DeepStreaks is quantified by the performance on the test data sets as well as using known near-Earth objects observed by ZTF. The system is deployed and adapted for usage within the ZTF Solar-System framework and has significantly reduced human involvement in the streak identification process, from several hours to typically under 10 minutes per day.