We report magnetization measurements under hydrostatic pressure up to approximately 1.6 GPa in the spin-1 quasi-one-dimensional frustrated antiferromagnets CsNiCl${}_3$, RbNiCl${}_3$, and CsNiBr${}_3$. Magnetic susceptibilities observed at various pressures are almost completely accounted for by combining the theoretical susceptibility for the spin-1 antiferromagnetic Heisenberg chain, calculated using the quantum Monte Carlo method, with the mean field arising from two interchain exchange interactions, which are an order of magnitude larger than the single interchain exchange interaction used in the conventional model. The antiferromagnetic intrachain $J_0$ exchange interaction increases significantly with pressure in all three. The single-ion anisotropy $D$ of the easy-axis type increases significantly with pressure in CsNiCl${}_3$ and RbNiCl${}_3$, while not in CsNiBr${}_3$. This significant change in magnetic interactions leads to large shifts in phase boundaries. One-dimensionality increases with increasing pressure in CsNiCl${}_3$, but not in RbNiCl${}_3$ and CsNiBr${}_3$.