AbstractHfTe₃ single crystal undergoes a charge-density-wave (CDW) transition at T_CDW = 93 K without the appearance of superconductivity (SC) down to 50 mK at ambient pressure. Here, we determined its CDW vector q = 0.91(1) a* + 0.27(1) c* via low-temperature transmission electron microscope and then performed comprehensive high-pressure transport measurements along three major crystallographic axes. Our results indicate that the superconducting pairing starts to occur within the quasi-one-dimensional (Q1D) -Te2-Te3- chain at 4–5 K but the phase coherence between the superconducting chains cannot be realized along either the b- or c-axis down to at least 1.4 K, giving rise to an extremely anisotropic SC rarely seen in real materials. We have discussed the prominent Q1D SC in pressurized HfTe₃ in terms of the anisotropic Fermi surfaces arising from the unidirectional Te-5p_x electronic states and the local pairs formed along the -Te2-Te3- chains based on the first-principles electronic structure calculations.