Brown algae are a source of hydrothermal extracts that can serve as an effective raw material for instant beverages. This article offers new formulations of functional instant beverages made of concentrated fruit juices and algal extracts of Saccharina japonica and Sargassum miyabei Yendo. The research objective was to define their bioactive and antioxidant profiles. The research featured S. miyabei Yendo and S. japonica brown algae from the Far East of Russia, their dry hydrothermal extracts, and instant drinks based on these extracts combined with concentrated juices of cranberry, sea buckthorn, and chokeberry. The list of methods included spectrophotometry, high-performance liquid chromatography, and gas chromatography. The hydrothermal algal extracts of S. miyabei and S. japonica were rich in fucoidan, phenolic compounds, and iodine. The new instant beverages underwent a sensory evaluation. They contained iodine, phenolic compounds, vitamins (ascorbic acid), fucoidan, pectin, flavonoids, anthocyanins, catechins, carotenoids, and tocopherols. All the samples could be classified as functional, but the best antiradical properties belonged to the sample with black chokeberry juice and S. miyabei. The new functional instant beverages had a high radical-binding activity, which reached 96.3%. One portion (200 mL) covered 27–30% of the recommended daily intake of iodine and 22–50% of vitamin C. The obtained results prove that instant beverages made of S. japonica and S. miyabei Yendo can be used as functional products.