In current online economic dispatch problems of power systems, the cost function of each generation unit typically only includes time-varying generation costs, which does not account for the costs associated with current carbon emission market transactions and high-frequency communications. These factors need to be considered. Additionally, considering the lag in obtaining information about cost coefficients, feedback delay should be considered as a factor for the online optimization process of economic dispatch. This research addresses the challenge of online economic dispatch in the presence of delayed feedback and carbon emission expenses, proposing a decentralized, event-initiated algorithm for online optimization. To address these challenges, this paper proposes a decentralized, event-triggered online optimization algorithm tailored for Agentic AI in smart energy systems operating over an edge–cloud continuum. In each local optimization iteration, each node in the power system can access its local objective function with a delayed time sequence and updates its local decision-making behavior online based on information from multiple neighboring nodes, controlled by an event-triggering mechanism, to minimize the global cumulative generation cost and carbon emission cost. The research results show that, under the assumption that the time-varying balanced undirected communication topology remains connected, the designed online optimization algorithm ensures that the upper bound of static network regret grows sub-linearly, fundamentally related to feedback delays and event-triggering thresholds and scaled as O ( T ) . This work provides a foundational paradigm for Agentic AI in smart energy systems, bridging distributed optimization theory with practical constraints of modern edge-cloud infrastructures.