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We construct an emulator for the halo mass function over group and cluster mass scales for a range of cosmologies, including the effects of dynamical dark energy and massive neutrinos. The emulator is based on the recently completed Mira-Titan Universe suite of cosmological $N$-body simulations. The main set of simulations spans 111 cosmological models with 2.1 Gpc boxes. We extract halo catalogs in the redshift range $z=[0.0, 2.0]$ and for masses $M_{200\mathrm{c}}\geq 10^{13}M_\odot/h$. The emulator covers an 8-dimensional hypercube spanned by {$Ω_\mathrm{m}h^2$, $Ω_\mathrm{b}h^2$, $Ω_νh^2$, $σ_8$, $h$, $n_s$, $w_0$, $w_a$}; spatial flatness is assumed. We obtain smooth halo mass functions by fitting piecewise second-order polynomials to the halo catalogs and employ Gaussian process regression to construct the emulator while keeping track of the statistical noise in the input halo catalogs and uncertainties in the regression process. For redshifts $z\lesssim1$, the typical emulator precision is better than $2\%$ for $10^{13}-10^{14} M_\odot/h$ and $<10\%$ for $M\simeq 10^{15}M_\odot/h$. For comparison, fitting functions using the traditional universal form for the halo mass function can be biased at up to 30\% at $M\simeq 10^{14}M_\odot/h$ for $z=0$. Our emulator is publicly available at \url{https://github.com/SebastianBocquet/MiraTitanHMFemulator}.