Quantitative photoacoustic imaging (QPAI) is a hybrid imaging technique aimed at reconstructing optical parameters from photoacoustic signals detected around the biological tissues. The recovery of optical parameters is a nonlinear, ill-posed inverse problem which is usually solved by iterative optimization methods based on the error minimization strategy. Most of the iterative algorithms are empirical and computationally expensive, leading to inadequate performance in practical application. In this work, we propose a deep learning-based QPAI approach to efficiently recover the optical absorption coefficient of biological tissues from the reconstructed result of initial pressure. The method involves a U-Net architecture based on the fully convolutional neural network. The Monte Carlo simulation with the wide-field illumination has been used to generate simulation data for the network training. The feasibility of the proposed method was demonstrated through numerical simulations, and its applicability to quantitatively reconstruct the distribution of optical absorption in the practical situation is further verified in phantom experiments. High image performance of this method in accuracy, efficiency and fidelity from both simulated and experimental results, suggests the enormous potential in biomedical applications in the future.