Linear-array-based photoacoustic tomography has shown broad applications in biomedical research and preclinical imaging. However, the elevational resolution of a linear array is fundamentally limited due to the weak cylindrical focus of the transducer element. While several methods have been proposed to address this issue, they have all handled the problem in a less time-efficient way. In this work, we propose to improve the elevational resolution of a linear array through Deep-E, a fully dense neural network based on U-net. Deep-E exhibits high computational efficiency by converting the three-dimensional problem into a two-dimension problem: it focused on training a model to enhance the resolution along elevational direction by only using the 2D slices in the axial and elevational plane and thereby reducing the computational burden in simulation and training. We demonstrated the efficacy of Deep-E using various datasets, including simulation, phantom, and human subject results. We found that Deep-E could improve elevational resolution by at least four times and recover the object's true size. We envision that Deep-E will have a significant impact in linear-array-based photoacoustic imaging studies by providing high-speed and high-resolution image enhancement.