Observables in the $D^0-\bar{D}^0$ mixing can be theoretically analyzed by the operator product expansion (OPE), in which $1/m_c$ is regarded as an expansion parameter. Since the contributions of four-quark operators are strongly suppressed by the Glashow-Iliopoulos-Maiani (GIM) mechanism, the order of magnitude of the width difference is still not reproduced in the OPE analysis. In view of this issue, quark-hadron duality, an assumption that is tacitly made in the OPE, is studied for the $D^0-\bar{D}^0$ mixing. In particular, the exclusive width difference and the inclusive counterpart can be compared within the 't Hooft model, two-dimensional QCD in the large-$N_c$ limit. It is shown that the order of magnitude of the exclusive width difference is enhanced relative to the 4D-like inclusive contributions of the four-quark operators, that is qualitatively consistent with the realistic observation of the $D^0-\bar{D}^0$ mixing.