Many examples of exposed giant dike swarms can be found where lateral magma flow has exceeded hundreds of kilometers. We show that massive magma flow into dikes can be established with only modest overpressure in a magma body if a large enough pathway opens at its boundary and gradual buildup of high tensile stress has occurred along the dike pathway prior to the onset of diking. This explains rapid initial magma flow rates, modeled up to about 7400 cubic meters per second into a dike ~15-kilometers long, which propagated under the town of Grindavík, Southwest Iceland, in November 2023. Such high flow rates provide insight into the formation of major dikes and imply a serious hazard potential for high-flow rate intrusions that propagate to the surface and transition into eruptions.