Recent approaches for video generation with camera control often create anchor videos (i.e., rendered videos that approximate desired camera motions) to guide diffusion models as a structured prior, by rendering from estimated point clouds following camera trajectories. However, errors in point cloud and camera trajectory estimation often lead to inaccurate anchor videos with higher training cost and low efficiency, as the model is forced to compensate for rendering misalignments. To address these limitations, we introduce EPiC, an efficient and precise camera control learning framework that constructs well-aligned training anchor videos without the need for camera pose or point cloud estimation. Concretely, we create highly precise anchor videos by masking source videos based on first-frame visibility, which ensures strong alignment, eliminates the need for camera/point cloud estimation, and thus can be readily applied to any in-the-wild video. Furthermore, we introduce Anchor-ControlNet, a lightweight module that integrates anchor video guidance in visible regions to pretrained video diffusion models, with less than 1% of additional parameters. EPiC achieves efficient training with substantially fewer parameters, training steps, and less data, and generalizes robustly to anchor videos made with point clouds at test time, enabling precise 3D-informed camera control. EPiC achieves SoTA performance on RealEstate10K and MiraData for I2V camera control task. Notably, EPiC also exhibits strong zero-shot generalization to video-to-video (V2V) scenarios.