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Optical absorption measurements characterize a wide variety of systems from atomic gases to in-vivo diagnostics of living organisms. Here we study the potential of non-classical techniques to reduce statistical noise below the shot-noise limit in absorption measurements with concomitant phase shifts imparted by a sample. We consider both cases where there is a known relationship between absorption and a phase shift, and where this relationship is unknown. For each case we derive the fundamental limit and provide a practical strategy to reduce statistical noise. Furthermore, we find an intuitive correspondence between measurements of absorption and of lossy phase shifts, which both show the same scope for precision enhancement. Our results demonstrate that non-classical techniques can aid real-world tasks with present-day laboratory techniques.