Machine learning models, especially deep neural networks have been shown to reveal membership information of inputs in the training data. Such membership inference attacks are a serious privacy concern, for example, patients providing medical records to build a model that detects HIV would not want their identity to be leaked. Further, we show that the attack accuracy amplifies when the model is used to predict samples that come from a different distribution than the training set, which is often the case in real world applications. Therefore, we propose the use of causal learning approaches where a model learns the causal relationship between the input features and the outcome. An ideal causal model is known to be invariant to the training distribution and hence generalizes well to shifts between samples from the same distribution and across different distributions. First, we prove that models learned using causal structure provide stronger differential privacy guarantees than associational models under reasonable assumptions. Next, we show that causal models trained on sufficiently large samples are robust to membership inference attacks across different distributions of datasets and those trained on smaller sample sizes always have lower attack accuracy than corresponding associational models. Finally, we confirm our theoretical claims with experimental evaluation on 4 datasets with moderately complex Bayesian networks and an image dataset of colored MNIST. We observe that neural network-based associational models exhibit upto 80 Our results confirm the value of the generalizability of causal models in reducing susceptibility to privacy attacks.