We use electromagnetically induced transparency (EIT) to probe the narrow electron-spin resonance of nitrogen-vacancy centers in diamond. Working with a multipass diamond chip at temperatures 6-30 K, the zero-phonon absorption line (637 nm) exhibits an optical depth of 6 and inhomogeneous linewidth of ~30 GHz FWHM. Simultaneous optical excitation at two frequencies separated by the ground-state zero-field splitting (2.88 GHz) reveals EIT resonances with a contrast exceeding 6% and FWHM down to 0.4 MHz. The resonances provide an all-optical probe of external electric and magnetic fields with a projected photon-shot-noise-limited sensitivity of 0.2 V/cm/√[Hz] and 0.1 nT/√[Hz], respectively. Operation of a prototype diamond-EIT magnetometer measures a noise floor of ~/<1 nT/√[Hz] for frequencies above 10 Hz and Allan deviation of 1.3±1.1 nT for 100 s intervals. The results demonstrate the potential of diamond-EIT devices for applications ranging from quantum-optical memory to precision measurement and tests of fundamental physics.