The experience of caregiver burden among family members of patients with advanced cancer is a common problem. The aim of this study was to determine whether the burden may be alleviated by means of a therapeutic approach based on self-chosen music. This randomised controlled trial (ClinicalTrials.gov, NCT04052074. Registered 9 August 2019) included 82 family caregivers of patients receiving home palliative care for advanced cancer. The intervention group (n = 41) listened to pre-recorded, self-chosen music for 30 min/day for seven consecutive days, while the control group (n = 41) listened to a recording of basic therapeutic education at the same frequency. The degree of burden was assessed by the Caregiver Strain Index (CSI), calculated before and after the seven-day intervention. According to this measure, caregiver burden fell significantly in the intervention group (CSI change: −0.56, SD 2.16) but increased in the control group (CSI change: +0.68, SD 1.47), with a significant group x moment interaction F(1, 80) = 9.30, p = 0.003, η2p = 0.11. These results suggest that, in the short term at least, the use of therapy based on self-chosen music alleviates the burden on family caregivers of palliative cancer patients. Moreover, this therapy is easy to administer at home and does not present any problems in practice.