Optically pumped deep-ultraviolet (DUV) lasing with low threshold was demonstrated from AlGaN-based multiple-quantum-well (MQW) heterostructures grown on sapphire substrates. The epitaxial layers were grown pseudomorphically by metalorganic chemical vapor deposition on (0001) sapphire substrates. Stimulated emission was observed at wavelengths of 256 nm and 249 nm with thresholds of 61 kW/cm² and 95 kW/cm² at room temperature, respectively. The thresholds are comparable to the reported state-of-the-art AlGaN-based MQW DUV lasers grown on bulk AlN substrates emitting at 266 nm. These low thresholds are attributed to the optimization of active region and waveguide layer as well as the use of high-quality AlN/sapphire templates. The stimulated emission above threshold was dominated by transverse-electric polarization. This work demonstrates the potential candidacy of sapphire substrates for DUV diode lasers.