A computationally effective method for decomposing r-fold tensor products of irreducible representations of U(N) in a basis-independent fashion is given. The multiplicity arising from the tensor decomposition is resolved with the eigenvalues of invariant operators chosen from the universal enveloping algebra generated by the infinitesimal operators of the dual (or complementary) representation. Shift operators which commute with the U(N) invariant operators, but not the dual invariant operators, are introduced to compute the eigenvectors and eigenvalues of the dual invariant operators algebraically. A three-fold tensor product of irreducible representations of SU(4) is decomposed to illustrate the power and generality of the method.