In this paper, we consider an optimal resource allocation problem over networked systems where the global cost function is a sum of local convex cost functions of the agents. This optimization problem is subject to an affine equality constraint which represents the demand that the agents should meet through weighted contribution of their cost variables. We propose a novel distributed continuous-time algorithm that solves the problem over networks with connected graph communication topology. We also present an extension of our algorithm to solve allocation problems subject to multiple affine constraints. We demonstrate our results through a numerical example.