Designing persuasive debunking messages poses significant challenges in addressing rumors during public health crises, especially when confronting diverse informational backgrounds. However, the underlying mechanisms through which the congruence between debunking messages and background information influences their persuasiveness remains unclear. Therefore, drawing upon the elaboration likelihood model (ELM) of persuasion theory, this study develops a theoretical model to explore how topic congruence and reputation congruence moderate the relationships between cognitive and affective appeals and debunking message persuasiveness. We collected 18,825 COVID-19 debunking posts and associated data from Sina Weibo. Employing linguistic psychoanalysis, machine learning algorithms, and econometric analysis, we operationalize key variables and subject our model to rigorous testing. The results indicated that cognitive appeals, including argument adequacy (β = 0.040, p < 0.01), evidence quality (β = 0.163, p < 0.01), expression certainty (β = 0.048, p < 0.01), media richness (β = 0.264, p < 0.01), and source credibility (β = 0.138, p < 0.01), as well as affective appeals, encompassing emotional pleasure (β = -0.020, p < 0.01), emotional arousal (β = 0.033, p < 0.01), and emotional dominance (β = 0.015, p < 0.01) significantly influenced debunking message persuasiveness through the central and peripheral cues of ELM, respectively. Moreover, topic congruence and reputation congruence exhibited differing moderating effects. Specifically, topic congruence amplified the impacts of cognitive appeals while weakening that of affective appeals on debunking message persuasiveness. Conversely, reputation congruence yielded inverse effects on the two types of appeals. These findings furnish invaluable insights for government authorities and social media platforms in comprehending the intricate mechanisms underlying debunking message persuasiveness, thereby facilitating the development of more persuasive strategies to counter rumors in public health emergencies.