We report the discovery of T dwarf companions to the nearby stars HN Peg (G0V, 18.4 pc, ~0.3 Gyr) and HD 3651 (K0V, 11.1 pc, ~7 Gyr). During an ongoing survey of 5x5 fields surrounding stars in the solar neighborhood with IRAC aboard the Spitzer Space Telescope, we identified these companions as candidate T dwarfs based on their mid-IR colors. Using near-IR spectra obtained with SpeX at the NASA IRTF, we confirm the presence of methane absorption that characterizes T dwarfs and measure spectral types of T2.5+/-0.5 and T7.5+/-0.5 for HN Peg B and HD 3651 B, respectively. By comparing our Spitzer data to images from 2MASS obtained several years earlier, we find that the proper motions of HN Peg B and HD 3651 B are consistent with those of the primaries, confirming their companionship. HN Peg B and HD 3651 B have angular separations of 43.2 and 42.9 from their primaries, which correspond to projected physical separations of 795 and 476 AU, respectively. A comparison of their luminosities to the values predicted by theoretical evolutionary models implies masses of 0.021+/-0.009 and 0.051+/-0.014 Msun for HN Peg B and HD 3651 B. In addition, the models imply an effective temperature for HN Peg B that is significantly lower than the values derived for other T dwarfs at similar spectral types, which is the same behavior reported by Metchev & Hillenbrand for the young late-L dwarf HD 203030 B. Thus, the temperature of the L/T transition appears to depend on surface gravity. Meanwhile, HD 3651 B is the first substellar companion directly imaged around a star that is known to harbor a close-in planet from RV surveys. The discovery of this companion supports the notion that the high eccentricities of close-in planets like the one near HD 3651 may be the result of perturbations by low-mass companions at wide separations.