We present the detection of three exoplanets orbiting the early M dwarf TOI-663 (TIC 54962195; V = 13.7 mag, J = 10.4 mag, R_★ = 0.512 ± 0.015 R_⊙, M_★ = 0.514 ± 0.012 M_⊙, d = 64 pc). TOI-663 b, c, and d, with respective radii of 2.27 ± 0.10 R_⊕, 2.26 ± 0.10 R_⊕, and 1.92 ± 0.13 R_⊕ and masses of 4.45 ± 0.65 M_⊕, 3.65 ± 0.97 M_⊕, and <5.2 M_⊕ at 99%, are located just above the radius valley that separates rocky and volatile-rich exoplanets. The planet candidates are identified in two TESS sectors and are validated with ground-based photometric follow-up, precise radial-velocity measurements, and high-resolution imaging. We used the software package juliet to jointly model the photometric and radial-velocity datasets, with Gaussian processes applied to correct for systematics. The three planets discovered in the TOI-663 system are low-mass mini-Neptunes with radii significantly larger than those of rocky analogs, implying that volatiles, such as water, must predominate. In addition to this internal structure analysis, we also performed a dynamical analysis that confirmed the stability of the system. The three exoplanets in the TOI-663 system, similarly to other sub-Neptunes orbiting M dwarfs, have been found to have lower densities than planets of similar sizes orbiting stars of different spectral types.