LaSb_22 exhibits a large, non-saturating, linear magnetoresistance at low temperatures, defying the expectations of Fermi liquid theory. This is thought to be caused by charge density wave order emerging below an abrupt, hysteretic anomaly in the resistivity at \sim∼ 355 K. We find that, under hydrostatic pressure, this anomaly becomes much more pronounced, develops strong hysteresis, and shifts rapidly towards lower temperatures. The anomaly is fully suppressed by only 6 kbar. Moreover, we observe a second transition anomaly at lower temperature, which likewise disappears under pressure. These findings are discussed in the context of a structural transition recently discovered in the sister material CeSb_22 and of density functional theory calculations, which indicate that the SmSb_22-type structure adopted by LaSb_22 at ambient conditions is unstable at moderate applied pressures.