In hyperspectral image (HSI) classification, convolutional neural networks (CNNs) have been widely employed and achieved promising performance. However, CNN-based methods face difficulties in achieving both accurate and efficient HSI classification due to their limited receptive fields and deep architectures. To alleviate these limitations, we propose an effective HSI classification network based on multi-head self-attention and spectral-coordinate attention (MSSCA). Specifically, we first reduce the redundant spectral information of HSI by using a point-wise convolution network (PCN) to enhance discriminability and robustness of the network. Then, we capture long-range dependencies among HSI pixels by introducing a modified multi-head self-attention (M-MHSA) model, which applies a down-sampling operation to alleviate the computing burden caused by the dot-product operation of MHSA. Furthermore, to enhance the performance of the proposed method, we introduce a lightweight spectral-coordinate attention fusion module. This module combines spectral attention (SA) and coordinate attention (CA) to enable the network to better weight the importance of useful bands and more accurately localize target objects. Importantly, our method achieves these improvements without increasing the complexity or computational cost of the network. To demonstrate the effectiveness of our proposed method, experiments were conducted on three classic HSI datasets: Indian Pines (IP), Pavia University (PU), and Salinas. The results show that our proposed method is highly competitive in terms of both efficiency and accuracy when compared to existing methods.