By training over large-scale datasets, zero-shot monocular depth estimation (MDE) methods show robust performance in the wild but often suffer from insufficiently precise details. Although recent diffusion-based MDE approaches exhibit appealing detail extraction ability, they still struggle in geometrically challenging scenes due to the difficulty of gaining robust geometric priors from diverse datasets. To leverage the complementary merits of both worlds, we propose BetterDepth to efficiently achieve geometrically correct affine-invariant MDE performance while capturing fine-grained details. Specifically, BetterDepth is a conditional diffusion-based refiner that takes the prediction from pre-trained MDE models as depth conditioning, in which the global depth context is well-captured, and iteratively refines details based on the input image. For the training of such a refiner, we propose global pre-alignment and local patch masking methods to ensure the faithfulness of BetterDepth to depth conditioning while learning to capture fine-grained scene details. By efficient training on small-scale synthetic datasets, BetterDepth achieves state-of-the-art zero-shot MDE performance on diverse public datasets and in-the-wild scenes. Moreover, BetterDepth can improve the performance of other MDE models in a plug-and-play manner without additional re-training.