Probing the mechanical properties of tissue on the microscale could aid in the identification of diseased tissues that are inadequately detected using palpation or current clinical imaging modalities, with potential to guide medical procedures such as the excision of breast tumours. Compression optical coherence elastography (OCE) maps tissue strain with microscale spatial resolution and can delineate microstructural features within breast tissues. However, without a measure of the locally applied stress, strain provides only a qualitative indication of mechanical properties. To overcome this limitation, we present quantitative micro-elastography, which combines compression OCE with a compliant stress sensor to image tissue elasticity. The sensor consists of a layer of translucent silicone with well-characterized stress-strain behaviour. The measured strain in the sensor is used to estimate the two-dimensional stress distribution applied to the sample surface. Elasticity is determined by dividing the stress by the strain in the sample. We show that quantification of elasticity can improve the ability of compression OCE to distinguish between tissues, thereby extending the potential for inter-sample comparison and longitudinal studies of tissue elasticity. We validate the technique using tissue-mimicking phantoms and demonstrate the ability to map elasticity of freshly excised malignant and benign human breast tissues.