Azithromycin (AZI) is one of the most commonly used macrolide antibiotics in children, but has the disadvantages of a heavy bitter taste and poor solubility. In order to solve these problems, hot-melt extrusion (HME) was used to prepare azithromycin amorphous solid dispersion. Preliminary selection of a polymer for HME was conducted by calculating Hansen solubility parameter to predict the miscibility of the drug and polymer. Eudragit® RL PO was chosen as the polymer due to its combination of taste-masking effect and dissolution. Moreover, the solubility was improved with this polymer. Design of experiments (DoE) was used to optimize the formulation and process, with screw speed, extrusion temperature, and drug percentage as independent variables, and content, dissolution, and extrudates diameter as dependent variables. The optimal extrusion parameters were obtained as follows: temperature—150 °C; screw speed—75 rpm; and drug percentage—25%. Differential scanning calorimetry (DSC) and Powder X-ray Diffraction (PXRD) studies of the powdered solid dispersions showed that the crystalline AZI transformed into the amorphous form. Fourier transform infrared spectroscopy (FTIR) results indicated that the formation of a hydrogen bond between AZI and the polymer led to the stabilization of AZI in its amorphous form. In conclusion, this work illustrated the importance of HME for the preparation of amorphous solid dispersion of AZI, which can solve the problems of bitterness and low solubility. It is also of great significance for the development of compliant pediatric AZI preparation.