  We have conducted a systematic study of X-ray emission from ultra-compact
dwarf (UCD) galaxies and extended star clusters (ESCs), based on archival {\sl
Chandra} observations. Among a sample of 511 UCDs and ESCs complied from the
literature, 17 X-ray counterparts with 0.5-8 keV luminosities above
$\sim$$5\times10^{36} {\rm~erg~s^{-1}}$ are identified, which are distributed
in eight early-type host galaxies. To facilitate comparison, we also identify
X-ray counterparts of 360 globular clusters (GCs) distributed in four of the
eight galaxies. The X-ray properties of the UCDs and ESCs are found to be
broadly similar to those of the GCs. The incidence rate of X-ray-detected UCDs
and ESCs, $(3.3\pm0.8)$\%, while lower than that of the X-ray-detected GCs
[($7.0\pm0.4)$\%], is substantially higher than expected from the field
populations of external galaxies. A stacking analysis of the individually
undetected UCDs/ESCs further reveals significant X-ray signals, which
corresponds to an equivalent 0.5-8 keV luminosity of $\sim$$4\times10^{35}
{\rm~erg~s^{-1}}$ per source. Taken together, these provide strong evidence
that the X-ray emission from UCDs and ESCs is dominated by low-mass X-ray
binaries having formed from stellar dynamical interactions, consistent with the
stellar populations in these dense systems being predominantly old. For the
most massive UCDs, there remains the possibility that a putative central
massive black hole gives rise to the observed X-ray emission.
