Microgreens are young, immature vegetables that contain higher concentrations of active compounds compared to mature vegetables and seeds. Radish microgreens are a good source of antioxidants, phenolic compounds, ascorbic acid, carotenoids, and anthocyanins. The production of microgreens is limited by their short shelf life due to higher dark respiration and accelerated senescence. The study was performed on three radish cultivars (Raphanus sativus L.): purple radish (cvP), red radish (cvR), and green radish (cvG). Radish microgreens were grown in chambers with controlled conditions (24 °C and a photoperiod of 16/8 h) under two types of artificial LED light (45 μmol m−2s−1): under white light (B:G:R) and a blue/red light combination (B:2R). The effect of the two types of light was examined on the 3rd, 7th, and 14th day after storage at a low temperature (+4 °C). The physiological status of the three cultivars of radish microgreens was examined by measuring the contents of total soluble phenolics, ascorbic acid, proteins, sugars, dry matter, anthocyanins, carotenoids, and chlorophyll as well as the total antioxidant activity. The results revealed that radish microgreens’ antioxidant capacity and phytochemical profile depend on the radish cultivar and on the type of LED light used for cultivation. It was shown that B:2R and red cultivar were most beneficial for the synthesis of most of the determined phytochemicals compared to B:G:R, or the purple and green cultivar, respectively. Storage at a low temperature in darkness slowed down most of the metabolic reactions during the first seven days, thus preserving most of the antioxidant activity.