Detecting ligand-protein interactions in living cells is a fundamental challenge in molecular biology and drug research. Proteome-wide profiling of thermal stability as a function of ligand concentration promises to tackle this challenge. However, current data analysis strategies use preset thresholds that can lead to suboptimal sensitivity/specificity tradeoffs and limited comparability across datasets. Here, we present a method based on statistical hypothesis testing on curves, which provides control of the false discovery rate. We apply it to several datasets probing epigenetic drugs and a metabolite. This leads us to detect off-target drug engagement, including the finding that the HDAC8 inhibitor PCI-34051 and its analog BRD-3811 bind to and inhibit leucine aminopeptidase 3. An implementation is available as an R package from Bioconductor ( https://bioconductor.org/packages/TPP2D ). We hope that our method will facilitate prioritizing targets from thermal profiling experiments.