The effect of extrusion cooking on bioactive compounds in third-generation snacks (TGSE) and microwave-expanded snacks (MWSE) prepared using black bean, blue maize, and chard (FBCS) was evaluated. FBCS was extruded at different moisture contents (MC; 22.2–35.7%), extrusion temperatures (ET; 102–142 °C), and screw speeds (SP; 96–171 rpm). Total anthocyanin content (TAC), contents of individual anthocyanins, viz., cyanidin-3-glucoside, malvidin-3-glucoside, pelargonidin-3-glucoside, pelargonidin-3-5-diglucoside, and delphinidin-3-glucoside chloride, total phenolic content (TPC), antioxidant activity (AA), and color parameters were determined. TAC and individual anthocyanin levels increased with the reduction in ET. ET and MC affected the chemical and color properties; increase in ET caused a significant reduction in TPC and AA. Microwave expansion reduced anthocyanin content and AA, and increased TPC. Extrusion under optimal conditions (29% MC, 111 rpm, and 120 °C) generated products with a high retention of functional compounds, with high TAC (41.81%) and TPC (28.23%). Experimental validation of optimized process parameters yielded an average error of 13.73% from the predicted contents of individual anthocyanins. Results suggest that the TGSE of FBCS obtained by combining extrusion and microwave expansion achieved significant retention of bioactive compounds having potential physiological benefits for humans.