Language models (LMs) may memorize personally identifiable information (PII) from training data, enabling adversaries to extract it during inference. Existing defense mechanisms such as differential privacy (DP) reduce this leakage, but incur large drops in utility. Based on a comprehensive study using circuit discovery to identify the computational circuits responsible PII leakage in LMs, we hypothesize that specific PII leakage circuits in LMs should be responsible for this behavior. Therefore, we propose PATCH (Privacy-Aware Targeted Circuit PatcHing), a novel approach that first identifies and subsequently directly edits PII circuits to reduce leakage. PATCH achieves better privacy-utility trade-off than existing defenses, e.g., reducing recall of PII leakage from LMs by up to 65%. Finally, PATCH can be combined with DP to reduce recall of residual leakage of an LM to as low as 0.01%. Our analysis shows that PII leakage circuits persist even after the application of existing defense mechanisms. In contrast, PATCH can effectively mitigate their impact.