Basaltic eruptions are the most common form of volcanism on Earth and planetary bodies. The low viscosity of basaltic magmas inhibits fragmentation, favouring effusive and lava-fountaining activity, yet highly explosive, hazardous basaltic eruptions occur. The processes that promote fragmentation of basaltic magma remain unclear and are subject to debate. Here we use a numerical conduit model to show that rapid magma ascent during explosive eruptions produces large undercooling. In situ experiments reveal that undercooling drives exceptionally rapid (~minutes) crystallisation, inducing a step-change in viscosity that triggers magma fragmentation. Experimentally produced textures are consistent with basaltic Plinian eruption products. We apply a numerical model to investigate basaltic magma fragmentation over a wide parameter space and find that all basaltic volcanoes have the potential to produce highly explosive eruptions. The critical requirements are initial magma temperatures lower than 1100 °C to reach a syn-eruptive crystal content of over 30 vol.%, and thus a magma viscosity around 105 Pa s, which our results suggest is the minimum viscosity required for the fragmentation of fast ascending basaltic magmas. These temperature, crystal content and viscosity requirements reveal how typically effusive basaltic volcanoes can produce unexpected highly explosive and hazardous eruptions.