Absorption spectroscopy is routinely used to characterise chemical and biological samples. For the state-of-the-art in laser absorption spectroscopy, precision is theoretically limited by shot-noise due to the fundamental Poisson-distribution of photon number in laser radiation. In practice, the shot-noise limit can only be achieved when all other sources of noise are eliminated. Here, we use wavelength-correlated and tuneable photon pairs to demonstrate how absorption spectroscopy can be performed with precision beyond the shot-noise limit and near the ultimate quantum limit by using the optimal probe for absorption measurement - single photons. We present a practically realisable scheme, which we characterise both the precision and accuracy of by measuring the response of a control feature. We demonstrate that the technique can successfully probe liquid samples and using two spectrally similar types of haemoglobin we show that obtaining a given precision in resolution requires fewer heralded single probe photons compared to using an idealised laser.