The performance of a deep neural network is heavily dependent on its architecture and various neural architecture search strategies have been developed for automated network architecture design. Recently, evolutionary neural architecture search (ENAS) has received increasing attention due to the attractive global optimization capability of evolutionary algorithms. However, ENAS suffers from extremely high computation costs because a large number of performance evaluations is usually required in evolutionary optimization and training deep neural networks is itself computationally very intensive. To address this issue, this paper proposes a new evolutionary framework for fast ENAS based on directed acyclic graph, in which parents are randomly sampled and trained on each mini-batch of training data. In addition, a node inheritance strategy is adopted to generate offspring individuals and their fitness is directly evaluated without training. To enhance the feature processing capability of the evolved neural networks, we also encode a channel attention mechanism in the search space. We evaluate the proposed algorithm on the widely used datasets, in comparison with 26 state-of-the-art peer algorithms. Our experimental results show the proposed algorithm is not only computationally much more efficiently, but also highly competitive in learning performance.