We demonstrate that the exact nonequilibrium steady state of the one-dimensional Heisenberg $XXZ$ spin chain driven by boundary Lindblad operators can be constructed explicitly with a matrix product ansatz for the nonequilibrium density matrix. For the isotropic Heisenberg chain, polarized at the boundaries in different directions with a non-zero twist angle, we calculate the exact magnetization profiles and magnetization currents. The in-plane steady-state magnetization profiles are harmonic functions with a frequency proportional to the twist angle. In-plane steady-state magnetization currents are subdiffusive and vanish as the boundary coupling strength increases, while the transverse current is diffusive and saturates as the coupling strength becomes large. The anisotropic chain exhibits spin helix states at special values of the anisotropy where the transverse current is independent of system size, even for non-integrable higher-spin chains.