  By extracting the transition amplitudes, we give the first lattice QCD
prediction of the two-body decay partial widths of the $1^{-+}$ charmoniumlike
hybrid $\eta_{c1}$. Given the calculated mass value $m_{\eta_{c1}}=4.329(36)$
GeV, the $\eta_{c1}$ decay is dominated by the open charm modes $D_1\bar{D}$,
$D^*\bar{D}$ and $D^*\bar{D}^*$ with partial widths of $258(133)$ MeV, $88(18)$
MeV and $150(118)$ MeV, respectively. The coupling of $\eta_{c1}$ to
$\chi_{c1}$ plus a flavor singlet pseudoscalar is not small, but
$\chi_{c1}\eta$ decay is suppressed by the small $\eta-\eta'$ mixing angle. The
partial width of $\eta_{c1}\to \eta_c\eta'$ is estimated to be around 1 MeV. We
suggest experiments to search for $\eta_{c1}$ in the $P$-wave $D^*\bar{D}$ and
$D^*\bar{D}^*$ systems. Especially, the polarization of $D^*\bar{D}^*$ can be
used to distinguish the $1^{-+}$ product (total spin $S=1$) from $1^{--}$
products ($S=0$).
