Employing various relativistic energy density functionals for nucleon-nucleon interactions, we investigate the impact of hyperons and $\Delta$ resonances on the frequencies of non-radial oscillations in neutron stars, where the universal coupling scheme is adopted for the $\Delta$-meson couplings. It is found that the inclusion of $\Delta$ resonances is essential for neutron stars to accommodate the recent mass and radius measurements of PSR J0030+0451, PSR J0740+6620, PSR J0437-4715, PSR J0614-3329, and HESS J1731-347. As $\Delta$ resonances start to emerge in neutron stars’ inner core, the $g$-mode oscillation energy becomes concentrated within the $\Delta$-admixed region, leading to a sharp increase in the $g$-mode frequency. We also examine the $f$-mode and $p$-mode oscillations and find that the impact of $\Delta$ resonances on these modes is less pronounced than the $g$-modes. The oscillation frequencies calculated in the Cowling approximation are then compared with those from full general relativity, confirming that the Cowling approximation introduces an error of approximately 20% for the $f$-modes and within 10% for the $g$-modes. The notable effect of $\Delta$ resonances on neutron stars’ $g$-mode frequencies holds important implications for probing the internal composition of neutron stars.