The Fermi-LAT unassociated sources represent some of the most enigmatic gamma-ray sources in the sky. Observations with the Swift-XRT and -UVOT telescopes have identified hundreds of likely X-ray and UV/optical counterparts in the uncertainty ellipses of the unassociated sources. In this work we present spectral fitting results for 205 possible X-ray/UV/optical counterparts to 4FGL unassociated targets. Assuming that the unassociated sources contain mostly pulsars and blazars, we develop a neural network classifier approach that applies gamma-ray, X-ray, and UV/optical spectral parameters to yield a descriptive classification of unassociated spectra into pulsars and blazars. From our primary sample of 174 Fermi sources with a single X-ray/UV/optical counterpart, we present 132 P_bzr_>0.99 likely blazars and 14 P_bzr_<0.01 likely pulsars, with 28 remaining ambiguous. These subsets of the unassociated sources suggest a systematic expansion to catalogs of gamma-ray pulsars and blazars. Compared to previous classification approaches our neural network classifier achieves significantly higher validation accuracy and returns more bifurcated Pbzr values, suggesting that multiwavelength analysis is a valuable tool for confident classification of Fermi unassociated sources.