Automated eyes-state classification from the EEG signals using nonlinear analysis tools is a new area of research. Based upon the theory of nonlinear analysis, recurrence plots (RPs) and recurrence quantification analysis (RQA) are of greater significance that help in understanding the chaotic and recurrence behavior of the dynamically occurring complex physiological signals. In our study, a novel method is proposed combining the RPs with the machine learning based algorithms for automated classification of EEG signals into eyes-open and eyes-close states. A huge dataset of 109 subjects has been acquired from the PhysioNet database. Each of the six RQA-based measures (recurrence rate, determinism, entropy, laminarity, trapping time, and longest vertical line)has been extracted from 64 EEG channels. Feature selection has been performed using genetic algorithm. The selected features have been averaged and combined to form a six-dimensional input vector which shows statistically significant differences (p < 0.01) between the two states. It is fed to different machine learning based algorithms such as logistic-regression, support vector machine, random forest, k-nearest neighbor, Gaussian naïve Bayes, and adaptive boosting. Logistic regression achieves the highest performance results in terms of accuracy, F1 score, precision, recall, and specificity of 97.27%, 97.17%, 98.26%, 96.36%, and 98.18%, respectively, with the least testing time of the model as 2.52 ms. Therefore, our method might be of greater significance in the development of the practical applications.