By exploiting the quantised nature of light, we demonstrate a sub-shot-noise scanning optical transmittance microscope. Our microscope demonstrates, with micron scale resolution, a factor of improvement in precision of 1.76(9) in transmittance estimation gained per probe photon relative to the theoretical model, a shot-noise-limited source of light, in an equivalent single-pass classical version of the same experiment using the same number of photons detected with a 90% efficient detector. This would allow us to observe photosensitive samples with nearly twice the precision, without sacrificing image resolution or increasing optical power to improve signal-to-noise ratio. Our setup uses correlated twin-beams produced by parametric down-conversion, and a hybrid detection scheme comprising photon-counting-based feed-forward and a highly efficient CCD camera. Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article s title, journal citation, and DOI.