Spectroscopy is an important tool for probing the properties of materials, chemicals, and biological samples. We design a practical transmitter-receiver system that exploits entanglement to achieve a provable quantum advantage over all spectroscopic schemes based on classical sources. To probe the absorption spectra, modeled as a pattern of transmissivities among different frequency modes, we employ broadband signal-idler pairs in two-mode squeezed vacuum states. At the receiver side, we apply photodetection after optical parametric amplification. Finally, we perform a maximum likelihood decision test on the measurement results, achieving an error probability orders of magnitude lower than the optimum classical systems in various examples, including "wine tasting" and "drug testing" where real molecules are considered. In detecting the presence of an absorption line, our quantum scheme achieves the optimum performance allowed by quantum mechanics. The quantum advantage in our system is robust against noise and loss, which makes near-term experimental demonstration possible.