  Structural phase transition can occur due to complex mechanisms other than
simple dynamical instability, especially when the parent and daughter structure
is of low dimension. This article reports such an inversion symmetry-breaking
structural phase transition in a quasi-1D compound (TaSe$_4$)$_3$I at T$_S\sim$
141~K studied by Raman spectroscopy. Our investigation of collective lattice
dynamics reveals three additional Raman active modes in the low-temperature
non-centrosymmetric structure. Two vibrational modes become Raman active due to
the absence of an inversion center, while the third mode is a soft phonon mode
resulting from the vibration of Ta atoms along the \{-Ta-Ta-\} chains.
Furthermore, the most intense Raman mode display Fano-shaped asymmetry,
inferred as the signature of strong electron-phonon coupling. The group theory
and symmetry analysis of Raman spectra confirm the displacive-first-order
nature of the structural transition. Therefore, our results establish
(TaSe$_4)_3$I as a model system with broken inversion symmetry and strong
electron-phonon coupling in the quasi-1D regime.
