We previously introduced a phase-aware variant of the non-negative matrix factorization (NMF) approach for audio source separation, which we call the 'Complex NMF (CNMF).' This approach makes it possible to realize NMF-like signal decompositions in the complex time-frequency domain. One limitation of the CNMF framework is that the divergence measure is limited to only the Euclidean distance. Some previous studies have revealed that for source separation tasks with NMF, the generalized Kullback-Leibler (KL) divergence tends to yield higher accuracy than when using other divergence measures. This motivated us to believe that CNMF could achieve even greater source separation accuracy if we could derive an algorithm for a KL divergence counterpart of CNMF. In this paper, we start by defining the notion of the 'dual' form of the CNMF formulation, derived from the original Euclidean CNMF, and show that a KL divergence counterpart of CNMF can be developed based on this dual formulation. We call this 'KL-CNMF'. We further derive a convergence-guaranteed iterative algorithm for KL-CNMF based on a majorization-minimization scheme. The source separation experiments revealed that the proposed KL-CNMF yielded higher accuracy than the Euclidean CNMF and NMF with varying divergences.