Transformer has been successfully used in practical applications, such as ChatGPT, due to its powerful advantages. However, users' input is leaked to the model provider during the service. With people's attention to privacy, privacy-preserving Transformer inference is on the demand of such services. Secure protocols for non-linear functions are crucial in privacy-preserving Transformer inference, which are not well studied. Thus, designing practical secure protocols for non-linear functions is hard but significant to model performance. In this work, we propose a framework \emph{East} to enable efficient and accurate secure Transformer inference. Firstly, we propose a new oblivious piecewise polynomial evaluation algorithm and apply it to the activation functions, which reduces the runtime and communication of GELU by over 1.5$\times$ and 2.5$\times$, compared to prior arts. Secondly, the secure protocols for softmax and layer normalization are carefully designed to faithfully maintain the desired functionality. Thirdly, several optimizations are conducted in detail to enhance the overall efficiency. We applied \emph{East} to BERT and the results show that the inference accuracy remains consistent with the plaintext inference without fine-tuning. Compared to Iron, we achieve about 1.8$\times$ lower communication within 1.2$\times$ lower runtime.