Turbulent high-energy astrophysical systems often feature asymmetric energy injection or driving: for instance, nonlinear interactions between Alfvén waves propagating from an accretion disk into its corona. Such systems-relativistic analogs of the solar wind-are "imbalanced": the energy fluxes parallel and anti-parallel to the large-scale magnetic field are unequal and the plasma therefore possesses net cross-helicity. In the past, numerical studies of imbalanced turbulence have focused on the magnetohydrodynamic regime. In the present study, we investigate externally-driven imbalanced turbulence in a collisionless, ultrarelativistically hot, magnetized pair plasma using three-dimensional particle-in-cell simulations. We find that a turbulent cascade forms for every value of imbalance covered by the simulations and that injected Poynting flux efficiently converts into net momentum of the plasma, a relativistic effect with implications for the launching of a disk wind. Surprisingly, particle acceleration remains efficient even for very imbalanced turbulence. These results characterize properties of imbalanced turbulence in a collisionless plasma and have ramifications for black hole accretion disk coronae, winds, and jets.