Solar X-ray jets are thought to be made by a burst of reconnection of closed magnetic field at the base of a jet with ambient open field(1,2). In the accepted version of the 'emerging-flux' model, such a reconnection occurs at a plasma current sheet between the open field and the emerging closed field, and also forms a localized X-ray brightening that is usually observed at the edge of the jet's base(1,3). Here we report high-resolution X-ray and extreme-ultraviolet observations of 20 randomly selected X-ray jets that form in coronal holes at the Sun's poles. In each jet, contrary to the emerging-flux model, a miniature version of the filament eruptions that initiate coronal mass ejections(4-7) drives the jet-producing reconnection. The X-ray bright point occurs by reconnection of the 'legs' of the minifilament-carrying erupting closed field, analogous to the formation of solar flares in larger-scale eruptions. Previous observations have found that some jets are driven by base-field eruptions(8-11), but only one such study, of only one jet, provisionally questioned the emerging-flux model(12). Our observations support the view that solar filament eruptions are formed by a fundamental explosive magnetic process that occurs on a vast range of scales, from the biggest mass ejections and flare eruptions down to X-ray jets, and perhaps even down to smaller jets that may power coronal heating(10,13,14). A similar scenario has previously been suggested, but was inferred from different observations and based on a different origin of the erupting minifilament(15).