Based on recently-taken and archival HARPS, FEROS, and HIRES radial velocities (RVs), we present evidence for a new planet orbiting the first ascent red giant star HD 33142 (with an improved mass estimate of M1.52 ± 0.03 M ⊙), already known to host two planets. We confirm the Jovian-mass planets HD 33142b and c, with periods of P b = 330.0 − 0.4 + 0.4 days and P c = 810.2 − 4.2 + 3.8 days and minimum dynamical masses of m b sin i = 1.26 − 0.05 + 0.05 M Jup and m c sin i = 0.89 − 0.05 + 0.06 MJup, respectively. Furthermore, our periodogram analysis of the precise RVs shows strong evidence for a short-period Doppler signal in the residuals of a two-planet Keplerian fit, which we interpret as a third, Saturn-mass planet with m d sin i = 0.20 − 0.03 + 0.02 M Jup in a close-in orbit with an orbital period of P d = 89.9 − 0.1 + 0.1 days. We study the dynamical behavior of the three-planet system configuration with an N-body integration scheme, finding it to be long-term stable with the planets alternating between low and moderate eccentricity episodes. We also perform N-body simulations, including stellar evolution and second-order dynamical effects such as planet-stellar tides and stellar mass loss on the way to the white dwarf phase. We find that planets HD 33142b, c, and d are likely to be engulfed near the tip of the red giant branch phase due to tidal migration. These results make the HD 33142 system an essential benchmark for planet population statistics of the multiple-planet systems found around evolved stars.