Superconductivity evolves as functions of pressure or doping from charge-ordered phases in a variety of strongly correlated systems, suggesting that there may be universal characteristics associated with the competition between superconductivity and charge order in these materials. We present an inelastic light (Raman) scattering study of the structural changes that precede the pressure-tuned charge-density-wave (CDW) to superconductor transition in one such system, ZrTe3. In certain phonon bands, we observe dramatic linewidth reductions that accompany CDW formation, indicating that these phonons couple strongly to the electronic degrees of freedom associated with the CDW. The same phonon bands, which represent internal vibrations of ZrTe3 prismatic chains, are suppressed at pressures above ~10 kbar, indicating a loss of long-range order within the chains, specifically amongst intrachain Zr-Te bonds. These results suggest a distinct structural mechanism for the observed pressure-induced suppression of CDW formation and provide insights into the origin of pressure-induced superconductivity in ZrTe3.