While human infants robustly discover their own causal efficacy, standard reinforcement learning agents remain brittle, as their reliance on correlation-based rewards fails in noisy, ecologically valid scenarios. To address this, we introduce the Causal Action Influence Score (CAIS), a novel intrinsic reward rooted in causal inference. CAIS quantifies an action's influence by measuring the 1-Wasserstein distance between the learned distribution of sensory outcomes conditional on that action, $p(h|a)$, and the baseline outcome distribution, $p(h)$. This divergence provides a robust reward that isolates the agent's causal impact from confounding environmental noise. We test our approach in a simulated infant-mobile environment where correlation-based perceptual rewards fail completely when the mobile is subjected to external forces. In stark contrast, CAIS enables the agent to filter this noise, identify its influence, and learn the correct policy. Furthermore, the high-quality predictive model learned for CAIS allows our agent, when augmented with a surprise signal, to successfully reproduce the "extinction burst" phenomenon. We conclude that explicitly inferring causality is a crucial mechanism for developing a robust sense of agency, offering a psychologically plausible framework for more adaptive autonomous systems.