We classify all irreducible highest-weight unitary modules over the non-compact real form u(p,q|n) of the general linear Lie superalgebra gl_(p+q|n). The classification is given by explicit necessary and sufficient conditions on the highest weights. Our approach combines the Howe duality for gl_(p+q|n) with a quadratic invariant of the maximal compact subalgebra. As consequences, we classify all irreducible lowest-weight unitary modules over u(p,q|n) via duality, and all irreducible unitary modules over u(n|q,p) via an isomorphism of Lie superalgebras.