Accurate hyperspectral image classification has been an important yet challenging task for years. With the recent success of deep learning in various tasks, 2-dimensional (2D)/3-dimensional (3D) convolutional neural networks (CNNs) have been exploited to capture spectral or spatial information in hyperspectral images. On the other hand, few approaches make use of both spectral and spatial information simultaneously, which is critical to accurate hyperspectral image classification. This paper presents a novel Synergistic Convolutional Neural Network (SyCNN) for accurate hyperspectral image classification. The SyCNN consists of a hybrid module that combines 2D and 3D CNNs in feature learning and a data interaction module that fuses spectral and spatial hyperspectral information. Additionally, it introduces a 3D attention mechanism before the fully-connected layer which helps filter out interfering features and information effectively. Extensive experiments over three public benchmarking datasets show that our proposed SyCNNs clearly outperform state-of-the-art techniques that use 2D/3D CNNs.