We present the projection operator method in combination with the Wigner-Racah calculus of the subalgebra Uq(su(2)) for calculation of Clebsch-Gordan coefficients (CGCs) of the quantum algebra Uq(su(3)). The key formulas of the method are couplings of the tensor and projection operators and also a tensor form for the projection operator of Uq(su(3)). We obtain a very compact general analytic formula for the Uq(su(3)) CGCs in terms of the Uq(su(2)) Wigner 3nj symbols.