Tuning a database system to achieve optimal performance on a given workload is a long-standing problem in the database community. A number of recent works have leveraged ML-based approaches to guide the sampling of large parameter spaces (hundreds of tuning knobs) in search for high performance configurations. Looking at Microsoft production services operating millions of databases, sample efficiency emerged as a crucial requirement to use tuners on diverse workloads. This motivates our investigation in LlamaTune, a tuner design that leverages domain knowledge to improve the sample efficiency of existing optimizers. LlamaTune employs an automated dimensionality reduction technique based on randomized projections, a biased-sampling approach to handle special values for certain knobs, and knob values bucketization, to reduce the size of the search space. LlamaTune compares favorably with the state-of-the-art optimizers across a diverse set of workloads. It identifies the best performing configurations with up to $11\times$ fewer workload runs, and reaching up to $21\%$ higher throughput. We also show that benefits from LlamaTune generalize across both BO-based and RL-based optimizers, as well as different DBMS versions. While the journey to perform database tuning at cloud-scale remains long, LlamaTune goes a long way in making automatic DBMS tuning practical at scale.