  We estimate the conventional astrophysical emission from dwarf spheroidal
satellite galaxies (dSphs) of the Milky Way, focusing on millisecond pulsars
(MSPs), and evaluate the potential for confusion with dark matter (DM)
annihilation signatures at GeV energies. In low-density stellar environments,
such as dSphs, the abundance of MSPs is expected to be proportional to stellar
mass. Accordingly, we construct the $\gamma$-ray luminosity function of MSPs in
the Milky Way disk, where $>90$ individual MSPs have been detected with the
$\textit{Fermi}$ Large Area Telescope (LAT), and scale this luminosity function
to the stellar masses of 30 dSphs to estimate the cumulative emission from
their MSP populations. We predict that MSPs within the highest stellar mass
dSphs, Fornax and Sculptor, produce a $\gamma$-ray flux $>500$~MeV of
$\sim10^{-11}$~ph~cm$^{-2}$~s$^{-1}$, which is a factor $\sim10$ below the
current LAT sensitivity at high Galactic latitudes. The MSP emission in
ultra-faint dSphs, including targets with the largest J-factors, is typically
several orders of magnitude lower, suggesting that these targets will remain
clean targets for indirect DM searches in the foreseeable future. For a DM
particle of mass 25~GeV annihilating to $b$ quarks at the thermal relic cross
section (consistent with DM interpretations of the Galactic Center excess), we
find that the expected $\gamma$-ray emission due to DM exceeds that of MSPs in
all of the target dSphs. Using the same Milky Way MSP population model, we also
estimate the Galactic foreground MSP coincidence probability along the same
sightlines to the dSphs.
