The accurate simulation and prediction of runoff in alpine glaciated watersheds is of increasing importance for the comprehensive management and utilization of water resources. In this study, long short-term memory (LSTM), a state-of-the-art artificial neural network algorithm, is applied to simulate the daily discharge of two data-sparse glaciated watersheds in the Tianshan Mountains in Central Asia. Two other classic machine learning methods, namely extreme gradient boosting (XGBoost) and support vector regression (SVR), along with a distributed hydrological model (Soil and Water Assessment Tool (SWAT) and an extended SWAT model (SWAT_Glacier) are also employed for comparison. This paper aims to provide an efficient and reliable method for simulating discharge in glaciated alpine regions that have insufficient observed meteorological data. The two typical basins in this study are the main tributaries (the Kumaric and Toxkan rivers) of the Aksu River in the south Tianshan Mountains, which are dominated by snow and glacier meltwater and precipitation. Our comparative analysis indicates that simulations from the LSTM shows the best agreement with the observations. The performance metrics Nash-Sutcliffe efficiency coefficient (NS) and correlation coefficient (R2) of LSTM are higher than 0.90 in both the training and testing periods in the Kumaric River Basin, and NS and R2 are also higher than 0.70 in the Toxkan River Basin. Compared to classic machine learning algorithms, LSTM shows significant advantages over most evaluating indices. XGBoost also has high NS value in the training period, but is prone to overfitting the discharge. Compared with the widely used hydrological models, LSTM has advantages in predicting accuracy, despite having fewer data inputs. Moreover, LSTM only requires meteorological data rather than physical characteristics of underlying data. As an extension of SWAT, the SWAT_Glacier model shows good adaptability in discharge simulation, outperforming the original SWAT model, but at the cost of increasing the complexity of the model. Compared with the oftentimes complex semi-distributed physical hydrological models, the LSTM method not only eliminates the tedious calibration process of hydrological parameters, but also significantly reduces the calculation time and costs. Overall, LSTM shows immense promise in dealing with scarce meteorological data in glaciated catchments.