We present Morpheus, a new model for generating pixel-level morphological classifications of astronomical sources. Morpheus leverages advances in deep learning to perform source detection, source segmentation, and morphological classification pixel-by-pixel via a semantic segmentation algorithm adopted from the field of computer vision. By utilizing morphological information about the flux of real astronomical sources during object detection, Morpheus shows resiliency to false-positive identifications of sources. We evaluate Morpheus by performing source detection, source segmentation, morphological classification on the Hubble Space Telescope data in the five CANDELS fields with a focus on the GOODS South field, and demonstrate a high completeness in recovering known GOODS South 3D-HST sources with H < 26 AB. We release the code publicly, provide online demonstrations, and present an interactive visualization of the Morpheus results in GOODS South.