Given a directed graph with non-negative costs and delays on each edge, the k edge-disjoint restricted shortest path problem (kRSP) is to find k edge-disjoint directed paths connecting a pair of distinct vertices s and t, with the aim of minimizing the total cost subject to a delay constraint. In this paper, we present an absolute approximation algorithm via the LP-rounding technique, which guarantees to find \(k-1\) edge-disjoint paths that strictly satisfy the delay constraint and have a total cost no more than that of an optimum solution. The key observation leading to our approach is that in any basic optimal solution of the linear programming relaxation for kRSP, the underlying graph composed of edges with fractional values forms exactly a cycle. We notably show that the cycle always contains a set of edges that, along with the integral edges from the LP, can compose a desired solution. Lastly, we present a method for rounding such a set of edges from this cycle to eventually obtain the desired solution.