High-throughput testing of drugs across molecular-characterised cell lines can identify candidate treatments and discover biomarkers. However, the cells' response to a drug is typically quantified by a summary statistic from a best-fit dose-response curve, whilst neglecting the uncertainty of the curve fit and the potential variability in the raw readouts. Here, we model the experimental variance using Gaussian Processes, and subsequently, leverage uncertainty estimates to identify associated biomarkers with a new Bayesian framework. Applied to in vitro screening data on 265 compounds across 1074 cancer cell lines, our models identified 24 clinically established drug-response biomarkers, and provided evidence for six novel biomarkers by accounting for association with low uncertainty. We validated our uncertainty estimates with an additional drug screen of 26 drugs, 10 cell lines with 8 to 9 replicates. Our method is applicable to any dose-response data without replicates, and improves biomarker discovery for precision medicine.