Existing pre-training methods for extractive Question Answering (QA) generate cloze-like queries different from natural questions in syntax structure, which could overfit pre-trained models to simple keyword matching. In order to address this problem, we propose a novel Momentum Contrastive pRe-training fOr queStion anSwering (MCROSS) method for extractive QA. Specifically, MCROSS introduces a momentum contrastive learning framework to align the answer probability between cloze-like and natural query-passage sample pairs. Hence, the pre-trained models can better transfer the knowledge learned in cloze-like samples to answering natural questions. Experimental results on three benchmarking QA datasets show that our method achieves noticeable improvement compared with all baselines in both supervised and zero-shot scenarios.