Fully ripe fruits and mature leaves of Elaeagnus angustifolia were harvested and analyzed by means of analytical and biological tests to better comprehend the chemical composition and therapeutic/nutraceutical potential of this plant. Fruits and leaves were dried and the obtained powders were analyzed to study their color character and (via headspace gas chromatography) describe the chemical profile. Subsequently, they were submitted to a chloroform–methanol extraction, to a hydroalcoholic extraction procedure assisted or not by microwaves, and to an extraction with supercritical CO2, assisted or not by ethanol as the co-solvent, to detect the polyphenolic and the volatile content. The resulting extracts were evaluated in terms of chlorophyll and carotenoid content, polyphenolic content, volatile fraction, total phenolic content, total flavonoid content, antioxidant activity, radical scavenging activity, and enzymatic inhibition activity. The results confirmed the correlation between the chemical composition and the high antioxidant potential of leaf extracts compared to the fruit extracts in terms of the phenolic and pigment content. A promising effect against tyrosinase emerged for all the extracts, suggesting a therapeutic/nutraceutical use for this plant. Conversely, the volatile content from both natural matrices was similar.