With the growing deployment of pre-trained models like Transformers on cloud platforms, privacy concerns about model parameters and inference data are intensifying. Existing Privacy-Preserving Transformer Inference (PPTI) frameworks face the "impossible trinity" of balancing privacy, efficiency, and performance: Secure Multi-Party Computation (SMPC)-based approaches ensure strong privacy but suffer from high computational overhead and performance losses; Conversely, permutation-based methods achieve near-plaintext efficiency and accuracy but compromise privacy by exposing sensitive model parameters and intermediate results. Bridging this gap with a single approach presents substantial challenges, motivating the introduction of CENTAUR, a groundbreaking PPTI framework that seamlessly integrates random permutations and SMPC to address the "impossible trinity". By designing efficient PPTI algorithms tailored to the structural properties of Transformer models, CENTAUR achieves an unprecedented balance among privacy, efficiency, and performance. Our experiments demonstrate CENTAUR's ability to resist diverse data reconstruction attacks, achieve plaintext-level inference accuracy, and boost inference speed by 5.0-30.4 times, unlocking new possibilities for secure and efficient AI deployment.