In this study, we calculate the mixing angles between the axial-vector mesons $D_{1(s1)}-D_{1(s1)}^{\prime }$ and $B_{1(s1)}-B_{1(s1)}^{\prime }$ using the QCD sum rules approach. Our results are ${\theta }_1=28.2\pm 0.6°$, ${\theta }_2=26.6\pm 0.6°$, ${\theta }_3=38.6\pm 0.1°$, and ${\theta }_4=38.5\pm 0.1°$. These values are in good agreement with the predictions of heavy quark effective theory, particularly for the mixing angle $\theta =35.3°$, and are compatible with several existing results in the literature. The predicted mixing angles can be tested through the analysis of semileptonic decays such as $B_c\to B_1\ell \nu$, $B_c\to B_{s1}^0\ell \nu$, $B_s\to D_{s1}\ell \nu$, and $B_s\to D_{s1}^{\prime }\ell \nu$, which can be investigated at experimental facilities such as LHCb and Belle II.