<p>This paper introduces the triangular fuzzy number (TFN)-valued gamma function and the T-Laplace transform, extending classical analytical operators to the TFN-valued functions. By leveraging a specialized component-wise structure for TFNs, we establish the fundamental properties and theoretical foundations of these transforms through rigorous proofs. The practical utility of the T-Laplace transform is demonstrated by solving fuzzy integro-differential equations within a T-electric circuit model. We demonstrate that for prescribed T-initial conditions, this framework maintains structural consistency throughout the transformation process. This enables the derivation of the TFN-valued analytical solution for the T-current, effectively modeling system uncertainty while preserving the characteristic TFN structure of the output.</p>