Five Hungarian sour cherry cultivars were studied to determine their anthocyanin contents and their possible inhibitory properties. The water and methanol soluble antioxidant capacities were separately assessed by photoluminescence showing values ranged from 3.4 μg mg−1 to 15.4 μg mg−1, respectively. The “VN1” variety (selected from “Csengődi csokros”) showed the highest antioxidant capacity. The anthocyanin content, measured by pH differential method or isolated by solid phase extraction, was the highest also in “VN1”. Correlation was found between the anthocyanin content and the high antioxidant capacity. The main anthocyanin components were cyanidin-3-O-rutinoside and cyanidin-3-O-glucoside. The presence of malvidin-3,5-O-diglycoside was verified by MALDI-TOF MS. Sour cherry extracts and selected anthocyanins inhibited the human salivary alpha-amylase catalyzed hydrolysis competitively. The lowest IC50 value, 55 μg mL−1 or 80 μM, was measured for malvidin-3,5-O-diglycoside, for which possible binding modes within the alpha-amylase active site could be investigated in silico using molecular docking and molecular dynamics.