Music is integral to the well-being of the elderly, and e-learning has become their primary avenue for music education. Grounded in flow theory, this study explores how the engagement of the elderly in music e-learning impacts their continuous music e-learning intentions and well-being. Analyzing valid responses from 357 elderly Chinese participants using SmartPLS, the study identifies three types of music e-learning engagement -- behavioral, emotional, and cognitive -- which positively influence their well-being but did not directly impact their continuous music e-learning intentions. In addition, flow experience mediated the effects of music e-learning engagement on well-being and continuous music e-learning intentions. The elderly's perceived information and communication technology (ICT) competence and perceived ICT autonomy positively moderated the relationship between music e-learning engagement and the flow experience. This study shed light on the positive role of music e-learning in enhancing the well-being of the elderly, offering valuable insights for music-learning platforms and governmental institutions to bolster the well-being of the elderly by supporting high-quality online music education and strengthening the ICT competence and autonomy of this demographic.