  We report the discovery of a bright X-ray source in the XMM-Newton and
Chandra X-ray Observatory (CXO) images of the unidentified Fermi-LAT source
3FGL J1016.5-6034. The XMM-Newton spectrum of the source is well fit by an
absorbed blackbody+power-law model with a temperature, $kT=0.20\pm0.02$ keV,
and photon index $\Gamma=1.8\pm0.1$. The CXO resolves the same source into a
point source (CXOU J101546.0-602939) and a surrounding compact nebula seen up
to about $30''$ from the point source. The CXO spectrum of the nebula can be
described by an absorbed power-law with $\Gamma=1.7\pm0.3$ and is partly
responsible for the non-thermal emission observed in the XMM-Newton spectrum.
The XMM-Newton images also reveal faint extended emission on arcminute scales.
These properties strongly suggest that the X-ray source and the accompanying
extended emission are a newly discovered young pulsar with a pulsar wind
nebula. We also analyze $\sim10$ years of Fermi-LAT data and find that the
improved LAT source localization is consistent with the position of CXOU
J101546.0-602939.
