As pre-trained models, like Transformers, are increasingly deployed on cloud platforms for inference services, the privacy concerns surrounding model parameters and inference data are becoming more acute. Current Privacy-Preserving Transformer Inference (PPTI) frameworks struggle with the "impossible trinity" of privacy, efficiency, and performance. For instance, Secure Multi-Party Computation (SMPC)-based solutions offer strong privacy guarantees but come with significant inference overhead and performance trade-offs. On the other hand, PPTI frameworks that use random permutations achieve inference efficiency close to that of plaintext and maintain accurate results but require exposing some model parameters and intermediate results, thereby risking substantial privacy breaches. Addressing this "impossible trinity" with a single technique proves challenging. To overcome this challenge, we propose Centaur, a novel hybrid PPTI framework. Unlike existing methods, Centaur protects model parameters with random permutations and inference data with SMPC, leveraging the structure of Transformer models. By designing a series of efficient privacy-preserving algorithms, Centaur leverages the strengths of both techniques to achieve a better balance between privacy, efficiency, and performance in PPTI. We comprehensively evaluate the effectiveness of Centaur on various types of Transformer models and datasets. Experimental results demonstrate that the privacy protection capabilities offered by Centaur can withstand various existing model inversion attack methods. In terms of performance and efficiency, Centaur not only maintains the same performance as plaintext inference but also improves inference speed by $5.0-30.4$ times.