Dendritic networks of nanoscopic grooves measuring 50–75 nm wide by <50 nm deep occur on the walls of vesicles in the glassy margins of mid-ocean ridge pillow basalts worldwide. Until now, their exact origin and significance have remained unclear. Here we document examples of such grooved patterns on vesicle walls in rocks from beneath the North Atlantic Ocean, and give a fluid mechanical explanation for how they formed. According to this model, individual nanogrooves represent frozen viscous fingers of magmatic fluid that were injected into a thin spheroidal shell of hot glass surrounding each vesicle. The driving mechanism for this process is provided by previous numerical predictions of tangential tensile stress around some vesicles in glassy rocks upon cooling through the glass transition. The self-assembling nature of the dendritic nanogrooves, their small size, and overall complexity in form, are interesting from the standpoint of exploring new applications in the field of nanotechnology. Replicating such structures in the laboratory would compete with state-of-the-art nanolithography techniques, both in terms of pattern complexity and size, which would be useful in the fabrication of a variety of grooved nanodevices. Dendritic nanogrooving in SiO2 glass might be employed in the manufacturing of integrated circuits.