We study the problem of energy-efficient online execution of a complex task on a server with variable processing speed. The task consists of a set of stochastic elementary jobs structured as a Directed Acyclic Graph (DAG), where each job's execution may reveal new information that influences future scheduling decisions. Our objective is to determine an online speed control policy that minimizes the expected energy consumption while ensuring that the task completes before a strict deadline. Leveraging tools from convex optimization, the optimality principle, and backward induction, we derive a structural characterization of the optimal policy. We find that this is linked to a set of second-order differential equations and exhibits a non-trivial form. Building on this result, we develop a discretization-based algorithm that efficiently approximates the optimal policy. The proposed algorithm is provably asymptotically exact in the discretization step and has computational complexity O(|A| + KN), where |A| and K denote
the number of edges and vertices (i.e., jobs) in the underlying DAG, and N is the discretization granularity. Our results offer a principled and computationally efficient solution framework for online execution of structured stochastic workloads under strict energy and timing constraints.