In recent years, there has been growing interest in bioactive plant compounds for their beneficial effects on health and for their potential in reducing the risk of developing certain diseases such as cancer, cardiovascular diseases, and neurodegenerative disorders. The extraction techniques conventionally used to obtain these phytocompounds, however, due to the use of toxic solvents and high temperatures, tend to be supplanted by innovative and unconventional techniques, in line with the demand for environmental and economic sustainability of new chemical processes. Among non-thermal technologies, cold plasma (CP), which has been successfully used for some years in the food industry as a treatment to improve food shelf life, seems to be one of the most promising solutions in green extraction processes. CP is characterized by its low environmental impact, low cost, and better extraction yield of phytochemicals, saving time, energy, and solvents compared with other classical extraction processes. In light of these considerations, this review aims to provide an overview of the potential and critical issues related to the use of CP in the extraction of phytochemicals, particularly polyphenols and essential oils. To review the current knowledge status and future insights of CP in this sector, a bibliometric study, providing quantitative information on the research activity based on the available published scientific literature, was carried out by the VOSviewer software (v. 1.6.18). Scientometric analysis has seen an increase in scientific studies over the past two years, underlining the growing interest of the scientific community in this natural substance extraction technique. The literature studies analyzed have shown that, in general, the use of CP was able to increase the yield of essential oil and polyphenols. Furthermore, the composition of the phytoextract obtained with CP would appear to be influenced by process parameters such as intensity (power and voltage), treatment time, and the working gas used. In general, the studies analyzed showed that the best yields in terms of total polyphenols and the antioxidant and antimicrobial properties of the phytoextracts were obtained using mild process conditions and nitrogen as the working gas. The use of CP as a non-conventional extraction technique is very recent, and further studies are needed to better understand the optimal process conditions to be adopted, and above all, in-depth studies are needed to better understand the mechanisms of plasma–plant matrix interaction to verify the possibility of any side reactions that could generate, in a highly oxidative environment, potentially hazardous substances, which would limit the exploitation of this technique at the industrial level.