Food packaging materials may be a potential source of contamination through the migration of components from the material into foodstuffs. Potential migrants can be known substances such as additives (e.g., plasticizers, stabilizers, antioxidants, etc.), monomers, and so on. However, they can also be unknown substances, which could be non-intentionally added substances (NIAS). In the present study, non-targeted analysis using mass spectrometry coupled to gas chromatography (GC-MS) for the identification of migrants in plastic and paper-based candy wrappers was performed. Samples were analyzed after extraction with acetonitrile. Numerous compounds including N-alkanes, phthalates, acetyl tributyl citrate, tributyl aconitate, bis(2-ethylhexyl) adipate, butylated hydroxytoluene, etc. were identified. Many of the compounds detected in plastic samples are not included in the positive list of the authorized substances. One non-intentionally added substance, 7,9-Di-tert-butyl-1-oxaspiro(4,5)deca-6-9-diene-2,8-dione, which has been reported as a degradation product of the antioxidant Irganox 1010, was found in several samples of both plastic and paper packaging. The proposed method was shown to be a useful approach for the identification of potential migrants in packaging samples. The toxicity of the compounds identified was estimated according to Cramer rules. Then, a second targeted analysis was also conducted in order to identify photoinitiators; among the analyzed compounds, only 2-hydroxybenzophenone was found in five samples.