We consider the boundary-driven interacting particle systems introduced in [FGK20a] related to the open non-compact Heisenberg model in one dimension. We show that a finite chain of $N$ sites connected at its ends to two reservoirs can be solved exactly, i.e. the non-equilibrium steady state has a closed-form expression for each $N$. The solution relies on probabilistic arguments and techniques inspired by integrable systems. It is obtained in two steps: i) the introduction of a dual absorbing process reducing the problem to a finite number of particles; ii) the solution of the dual dynamics exploiting a symmetry obtained from the Quantum Inverse Scattering Method. The exact solution allows to prove by a direct computation that, in the thermodynamic limit, the system approaches local equilibrium. A by-product of the solution is the algebraic construction of a direct mapping (a conjugation) between the generator of the non-equilibrium process and the generator of the associated reversible equilibrium process.