We propose a robust estimator for the tail index of Pareto-type distributions under random right-censoring, constructed within the minimum density power divergence (MDPD) framework and based on the Nelson--Aalen estimator of the cumulative hazard function. To our knowledge, this is the first application of the MDPD methodology to tail index estimation in the presence of random censoring. Under mild regularity conditions and within the weak censoring regime, the estimator is shown to be consistent and asymptotically normal. Its finite-sample performance is assessed through Monte Carlo simulations, revealing improved robustness--efficiency trade-offs compared to standard non-robust tail index estimators. Robustness is investigated under both pre-censoring and post-censoring contamination schemes. While pre-censoring contamination provides a meaningful framework for robustness assessment, post-censoring contamination directly alters the observable data and highlights the sensitivity of reconstruction-based approaches. The practical relevance of the method is illustrated using an insurance claims dataset with light censoring and fully observable extremes. An additional application to AIDS survival data is included for illustrative purposes, emphasizing the challenges encountered under stronger censoring.