Spatial resolution is a critical imaging parameter in magnetic resonance imaging (MRI). Acquiring high resolution MRI data usually takes long scanning time and would subject to motion artifacts due to hardware, physical, and physiological limitations. Single image super-resolution (SISR), especially that based on deep learning techniques, is an effective and promising alternative technique to improve the current spatial resolution of magnetic resonance (MR) images. However, the deeper network is more difficult to be effectively trained because the information is gradually weakened as the network deepens. This problem becomes more serious for medical images due to the degradation of training examples. In this paper, we present a novel channel splitting and serial fusion network (CSSFN) for single MR image super-resolution. Specifically, the proposed CSSFN network splits the hierarchical features into a series of subfeatures, which are then integrated together in a serial manner. Thus, the network becomes deeper and can deal with the subfeatures on different channels discriminatively. Besides, a dense global feature fusion (DGFF) is adopted to integrate the intermediate features, which further promotes the information flow in the network. Extensive experiments on several typical MR images show the superiority of our CSSFN model over other advanced SISR methods.