(1) Background: Tuberous sclerosis complex (TSC) mutations directly affect mTORC activity and, as a result, protein synthesis. In several cancer types, TSC mutation is part of the driver mutation panel. TSC mutations have been associated with mitochondrial dysfunction, tolerance to reactive oxygen species due to increased thioredoxin reductase (TrxR) enzyme activity, tolerance to endoplasmic reticulum (ER) stress, and apoptosis. The FDA-approved drug rapamycin is frequently used in clinical applications to inhibit protein synthesis in cancers. Recently, TrxR inhibitor auranofin has also been involved in clinical trials to investigate the anticancer efficacy of the combination treatment with rapamycin. We aimed to investigate the molecular background of the efficacy of such drug combinations in treating neoplasia modulated by TSC mutations. (2) Methods: TSC2 mutant and TSC2 wild-type (WT) cell lines were exposed to rapamycin and auranofin in either mono- or combination treatment. Mitochondrial membrane potential, TrxR enzyme activity, stress protein array, mRNA and protein levels were investigated via cell proliferation assay, electron microscopy, etc. (3) Results: Auranofin and rapamycin normalized mitochondrial membrane potential and reduced proliferation capacity of TSC2 mutant cells. Database analysis identified peroxiredoxin 5 (Prdx5) as the joint target of auranofin and rapamycin. The auranofin and the combination of the two drugs reduced Prdx5 levels. The combination treatment increased the expression of heat shock protein 70, a cellular ER stress marker. (4) Conclusions: After extensive analyses, Prdx5 was identified as a shared target of the two drugs. The decreased Prdx5 protein level and the inhibition of both TrxR and mTOR by rapamycin and auranofin in the combination treatment made ER stress-induced cell death possible in TSC2 mutant cells.