Recent Visual Question Answering (VQA) models have shown impressive performance on the VQA benchmark but remain sensitive to small linguistic variations in input questions. Existing approaches address this by augmenting the dataset with question paraphrases from visual question generation models or adversarial perturbations. These approaches use the combined data to learn an answer classifier by minimizing the standard cross-entropy loss. To more effectively leverage the augmented data, we build on the recent success in contrastive learning. We propose a novel training paradigm (ConCAT) that alternately optimizes cross-entropy and contrastive losses. The contrastive loss encourages representations to be robust to linguistic variations in questions while the cross-entropy loss preserves the discriminative power of the representations for answer classification. We find that alternately optimizing both losses is key to effective training. VQA models trained with ConCAT achieve higher consensus scores on the VQA-Rephrasings dataset as well as higher VQA accuracy on the VQA 2.0 dataset compared to existing approaches across a variety of data augmentation strategies.