In randomized clinical trials, improving efficiency and reducing bias due to chance imbalance in covariates among groups are always of considerable interest. The two purposes are often achieved by some type of covariate adjustment. In trials involving time-to-an-event, Kaplan-Meier and Nelson-Aalen estimators are the most popular nonparametric estimation of survival curves. However, these methods do not permit direct covariate adjustment, missing the important chance of improving efficiency and reducing bias. In this article, we propose robust, covariate adjusted analogues of the Nelson-Aalen and Kaplan-Meier estimators. The method is robust in that it does not require any additional modeling assumptions and hence the resulting estimators are again nonparametric. The robustness is achieved by taking advantage of the study design, i.e., treatments are randomized. Large-sample properties of the proposed estimators are developed, which show that the improvement in efficiency is guaranteed asymptotically. Simulation studies using reasonably small sample sizes further demonstrate the efficiency gain and the ability to reduce or remove bias resulted from chance imbalance to a large degree, e.g., more than 10-fold reduction in bias is achieved. Efficiency improvement and bias reduction are also illustrated by application to a cancer clinical trial. The proposed methods may help to resolve the tension between the need to make best use of data and the unwillingness to make additional assumptions in analyzing data from clinical trials.