Sparse-view Computed Tomography (CT) has the ability to reduce radiation dose and shorten the scan time, while the severe streak artifacts will compromise anatomical information. How to reconstruct high-quality images from sparsely sampled projections is a challenging ill-posed problem. In this context, we propose the unrolled Deep Residual Error iterAtive Minimization Network (DREAM-Net) based on a novel iterative reconstruction framework to synergize the merits of deep learning and iterative reconstruction. DREAM-Net performs constraints using deep neural networks in the projection domain, residual space, and image domain simultaneously, which is different from the routine practice in deep iterative reconstruction frameworks. First, a projection inpainting module completes the missing views to fully explore the latent relationship between projection data and reconstructed images. Then, the residual awareness module attempts to estimate the accurate residual image after transforming the projection error into the image space. Finally, the image refinement module learns a non-standard regularizer to further fine-tune the intermediate image. There is no need to empirically adjust the weights of different terms in DREAM-Net because the hyper-parameters are embedded implicitly in network modules. Qualitative and quantitative results have demonstrated the promising performance of DREAM-Net in artifact removal and structural fidelity.