The recent observation of high-harmonic generation from solids creates a new possibility for engineering fundamental strong-field processes by patterning the solid target with subwavelength nanostructures. All-dielectric metasurfaces exhibit high damage thresholds and strong enhancement of the driving field, making them attractive platforms to control high harmonics and other high-field processes at the nanoscale. Here we report enhanced non-perturbative high-harmonic emission from a Fano-resonant Si metasurface that possesses a classical analogue of electromagnetically induced transparency. The harmonic emission is enhanced by more than two orders of magnitude compared to unpatterned samples. The enhanced high harmonics are highly anisotropic with respect to the excitation polarization and are selective by the excitation wavelength due to its resonant features. By combining nanofabrication technology and ultrafast strong-field physics, our work paves the way for the design of new compact ultrafast photonic devices that operate under high intensities and at short wavelengths.