El Niño-Southern Oscillation (ENSO), which is one of the main drivers of Earth's inter-annual climate variability, often causes a wide range of climate anomalies, and the advance prediction of ENSO is always an important and challenging scientific issue. Since a unified and complete ENSO theory has yet to be established, people often use related indicators, such as the Niño 3.4 index and southern oscillation index (SOI), to predict the development trends of ENSO through appropriate numerical simulation models. However, because the ENSO phenomenon is a highly complex and dynamic model and the Niño 3.4 index and SOI mix many low- and high-frequency components, the prediction accuracy of current popular numerical prediction methods is not high. Therefore, this paper proposed the ensemble empirical mode decomposition-temporal convolutional network (EEMD-TCN) hybrid approach, which decomposes the highly variable Niño 3.4 index and SOI into relatively flat subcomponents and then uses the TCN model to predict each subcomponent in advance, finally combining the sub-prediction results to obtain the final ENSO prediction results. Niño 3.4 index and SOI reanalysis data from 1871 to 1973 were used for model training, and the data for 1984-2019 were predicted 1 month, 3 months, 6 months, and 12 months in advance. The results show that the accuracy of the 1-month-lead Niño 3.4 index prediction was the highest, the 12-month-lead SOI prediction was the slowest, and the correlation coefficient between the worst SOI prediction result and the actual value reached 0.6406. Furthermore, the overall prediction accuracy on the Niño 3.4 index was better than that on the SOI, which may have occurred because the SOI contains too many high-frequency components, making prediction difficult. The results of comparative experiments with the TCN, LSTM, and EEMD-LSTM methods showed that the EEMD-TCN provides the best overall prediction of both the Niño 3.4 index and SOI in the 1-, 3-, 6-, and 12-month-lead predictions among all the methods considered. This result means that the TCN approach performs well in the advance prediction of ENSO and will be of great guiding significance in studying it.