We present the first major release of data from the SAMI Galaxy Survey. This data release focuses on the emission-line physics of galaxies. Data Release One includes data for 772 galaxies, about 20 per cent of the full survey. Galaxies included have the redshift range 0.004 < z < 0.092, a large mass range (7.6 < logM*/M⊙ < 11.6), and star formation rates of ~10-4 to ~101M⊙ yr-1. For each galaxy, we include two spectral cubes and a set of spatially resolved 2D maps: single- and multi-component emission-line fits (with dust-extinction corrections for strong lines), local dust extinction, and star formation rate. Calibration of the fibre throughputs, fluxes, and differential atmospheric refraction has been improved over the Early Data Release. The data have average spatial resolution of 2.16 arcsec (full width at half-maximum) over the 15 arcsec diameter field of view and spectral (kinematic) resolution of R = 4263 (σ = 30 km s-1) around Ha. The relative flux calibration is better than 5 per cent, and absolute flux calibration has an rms of 10 per cent. The data are presented online through the Australian Astronomical Observatory's Data Central.