The plant species Opopanax hispidus is known in traditional Turkish medicine and nutrition. In addition to all the advantages that are known by the traditions of old generations, O. hispidus has been insufficiently examined, and its potential could be used in treatment in modern medicine and diet. In this paper, O. hispidus extracts obtained by conventional (maceration (MAC), and Soxhlet (SOX)) and contemporary (accelerated solvent extraction (ASE), homogenizer assisted extraction (HAE), microwave-assisted extraction (MAE), ultrasound-assisted extraction (UAE), supercritical fluid extraction (SFE)) methods were examined in terms of their chemical composition and biological potential. The extract obtained by HAE extraction had the highest content of total phenolics and flavonoids (36.98 mg GAE/g and 15.04 mg RE/g, respectively). The highest antioxidant activity was recorded for HAE and ASE extracts. O. hispidus extracts achieved inhibitory activity against acetylcholinesterase (4.12–6.44 mg GALAE/g), tyrosinase (44.78–76.72 mg KAE/g), α-amylase (0.35–0.46 mmol ACAE/g), and α-glucosidase (1.67–1.77 mmol ACAE/g) enzymes. LC-MS technique was used for the quantitative analysis of the extracts, allowing the quantification of chlorogenic acid, neochlorogenic acid, and p-coumaric acid, rutin, luteolin, and kaempferol 3-O-glucoside as the most abundant phenolic components. Results suggested that the green extraction techniques for obtaining extracts that are rich in bioactive compounds have advantages compared to conventional extraction. Therefore, these extracts can be used as antioxidants, neuroprotective, antityrosinase, and antidiabetic agents while respecting the principles of sustainable development and green chemistry.