Very Large Telescope/UVES spectroscopic and Transiting Exoplanet Survey Satellite (TESS) photometric observations for WASP 0346-21 allow the direct determination of its physical properties, along with the detection of a circumbinary object and oscillating signals. The high-resolution spectra yielded the radial velocities of all three stars and the atmospheric parameters of T eff,A = 7225 ± 42 K, [M/H] = 0.30 ± 0.03 dex, and v A sini = 78 ± 5 km s−1 of the primary component. The combined analysis of these observations resulted in the fundamental parameters of the eclipsing components and the third light of l 3 = 0.043 ± 0.004, which is consistent with the light contribution of the tertiary star observed in the echelle spectra. WASP 0346-21 A resides within the overlapping main-sequence domain of δ Sct and γ Dor variables, while the secondary component of M B = 0.185 ± 0.013 M ⊙, R B = 0.308 ± 0.023 R ⊙, T eff,B = 10,655 ± 146 K, and L B = 1.09 ± 0.17 L ⊙ matches well with the low-mass white dwarf (WD) model for Z = 0.01, corresponding to the thick disk population classified by the Galactic kinematics. Multifrequency analyses were performed on the residual TESS data after removing the binarity effects. The low frequencies around 26.348 day−1 and 17.683 day−1 are δ Sct pulsations originating from WASP 0346-21 A, and the high frequencies of 97.996 day−1 and 90.460 day−1 are considered to be extremely low-mass WD oscillations. These results demonstrate that WASP 0346-21 is a hierarchical triple system, consisting of an EL CVn binary with multiperiodic pulsations in each component and a distant outer tertiary.