(1) Background: Hypertensive patients with obstructive sleep apnea (OSA) are at high risk for cardiovascular diseases (CVDs), and the utility of aspirin for primary cardiovascular prevention in this population remains uncertain. (2) Methods: In this retrospective cohort study using data from the Urumchi Hypertension Database (UHDATA), hypertensive patients older than 18 years old with a first-time diagnosis of OSA were divided into three groups depending on aspirin history. Major adverse cardiac and cerebrovascular events (MACCE) were the primary outcome. Secondary outcomes included MACCE components, ischemic events, cardiac events, cerebrovascular events, and gastrointestinal bleeding risk. The inverse probability of treatment weighting (IPTW) method was used to balance the confounding factors among the groups, and the Cox proportional hazards model was used to calculate the hazard ratio (HR) and 95% confidence interval (CI). (3) Results: In persistent aspirin users, the risk of MACCE events (HR 2.11, 95%CI 1.23–3.63), ischemic events (HR 2.58, 95%CI 1.42–4.69), cerebrovascular events (HR 2.55, 95%CI 1.44–4.51), and non-fatal cerebral infarction (HR 3.14, 95%CI 1.69–5.84) was significantly elevated. (4) Conclusions: Continuous aspirin use increases the incidence of cardiovascular adverse events in hypertensive patients with OSA receiving aspirin for primary prevention of cardiovascular disease.