We report a small-angle neutron scattering study of the helical and skyrmion lattice order in single-crystal Cu₂OSeO₃ under quasihydrostatic helium gas pressures up to 5 kbar. By using helium gas as the pressure-transmitting medium (PTM) we ensure pressure application with improved hydrostaticity at cryogenic temperatures compared with previous reports where liquid PTMs were used. For 5-kbar He gas pressure we observe modest changes of the ambient pressure phase diagram; the critical temperature T_c changes by +2.8(2)%, while in the low-T limit the helical propagation vector |q| changes by −0.5(2)%, the lower critical field Hc1 changes by +2.5(1.0)%, and the upper critical field H_c2 remains unchanged within uncertainty. The skyrmion phase also changes little under pressure; its largest T extent varies from T_c−2.5(5) K at ambient pressure to T_c−3.0(5) K at 5 kbar, and its location in the phase diagram follows the pressure-driven shift of T_c. The weak pressure dependences of the critical magnetic fields and skyrmion phase contrast strongly with much stronger pressure-driven changes reported from previous quasihydrostatic pressure studies. Taking into account the present results and those of other uniaxial pressure data, we suggest that the results of previous quasihydrostatic pressure studies were influenced by inadvertent directional stress pressure components. Overall, our study represents a high-pressure study of the chiral magnetism in Cu₂OSeO₃ under the most hydrostatic high-pressure conditions to date and serves also as a salient reminder of the sensitivity of chiral magnets to deviations from hydrostaticity in quasihydrostatic high-pressure studies.