Time series prediction with deep learning methods, especially long short-term memory neural networks (LSTMs), have scored significant achievements in recent years. Despite the fact that the LSTMs can help to capture long-term dependencies, its ability to pay different degree of attention on sub-window feature within multiple time-steps is insufficient. To address this issue, an evolutionary attention-based LSTM training with competitive random search is proposed for multivariate time series prediction. By transferring shared parameters, an evolutionary attention learning approach is introduced to the LSTMs model. Thus, like that for biological evolution, the pattern for importance-based attention sampling can be confirmed during temporal relationship mining. To refrain from being trapped into partial optimization like traditional gradient-based methods, an evolutionary computation inspired competitive random search method is proposed, which can well configure the parameters in the attention layer. Experimental results have illustrated that the proposed model can achieve competetive prediction performance compared with other baseline methods.