We report the confirmation of three exoplanets transiting TOI-4010 (TIC-352682207), a metal-rich K dwarf observed by TESS in Sectors 24, 25, 52, and 58. We confirm these planets with HARPS-N radial velocity observations and measure their masses with 8 - 12% precision. TOI-4010 b is a sub-Neptune (\(P = 1.3\) days, \(R_{p} = 3.02_{-0.08}^{+0.08}~R_{\oplus}\), \(M_{p} = 11.00_{-1.27}^{+1.29}~M_{\oplus}\)) in the hot Neptune desert, and is one of the few such planets with known companions. Meanwhile, TOI-4010 c (\(P = 5.4\) days, \(R_{p} = 5.93_{-0.12}^{+0.11}~R_{\oplus}\), \(M_{p} = 20.31_{-2.11}^{+2.13}~M_{\oplus}\)) and TOI-4010 d (\(P = 14.7\) days, \(R_{p} = 6.18_{-0.14}^{+0.15}~R_{\oplus}\), \(M_{p} = 38.15_{-3.22}^{+3.27}~M_{\oplus}\)) are similarly-sized sub-Saturns on short-period orbits. Radial velocity observations also reveal a super-Jupiter-mass companion called TOI-4010 e in a long-period, eccentric orbit (\(P \sim 762\) days and \(e \sim 0.26\) based on available observations). TOI-4010 is one of the few systems with multiple short-period sub-Saturns to be discovered so far.