Crystal-field (CF) effects on the rare-earth (RE) ions in ferrimagnetic intermetallics NdCo$_5$ and TbCo$_5$ are evaluated using an ab initio density functional + dynamical mean-field theory approach in conjunction with a quasi-atomic approximation for on-site electronic correlations on the localized 4$f$ shell. The study reveals an important role of the high-order sectoral harmonic component of the CF in the magnetism of RECo$_5$ intermetallics. An unexpectedly large value is computed in the both systems for the corresponding crystal-field parameter (CFP) $A_6^6 langle r^6 rangle$, far beyond what one would expect from only electrostatic contributions. It allows solving the enigma of the non-saturation of zero-temperature Nd magnetic moments in NdCo$_5$ along its easy axis in the Co exchange field. This unsaturated state had been previously found out from magnetization distribution probed by polarised neutron elastic scattering but had so far remained theoretically unexplained. The easy plane magnetic anisotropy of Nd in NdCo$_5$ is strongly enhanced by the large value of $A_6^6langle r^6 rangle$. Counter-intuitively, the polar dependence of anisotropy energy within the easy plane remains rather small. The easy plane magnetic anisotropy of Nd is reinforced up to high temperatures, which is explained through $J$-mixing effects. The calculated ab initio anisotropy constants of NdCo$_5$ and their temperature dependence are in quantitative agreement with experiment. Unlike NdCo$_5$, the $A_6^6 langle r^6 rangle$ CFP has negligible effects on the Tb magnetism in TbCo$_5$ suggesting that its impact on the RE magnetism is ion-specific across the RECo$_5$ series. The origin of its large value is the hybridization of RE and Co states in a hexagonally coordinated local environment of the RE ion in RECo$_5$ intermetallics.
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