One of the important non-linear data structures in Computer Science is graph. Most of the real life network, be it a road transportation network, or airlines network or a communication network etc., cannot be exactly transformed into a graph model, but into a Multigraphs model. The Multigraph is a topological generalization of the graph where multiple links (or edges/arcs) may exist between two nodes unlike in graph. The existing algorithms to extract the neutrosophic shortest path in a graph cannot be applied to a Multigraphs. In this paper a method is developed to extract the neutrosophic shortest path in a directed Multigraph and then the corresponding algorithm is designed. The classical Dijkstra’s algorithm is applicable to graphs only where all the link weights are crisp, but we borrow this concept to apply to Multigraphs where the weights of the links are neutrosophic numbers (NNs). This new method may be useful in many application areas of computer science, communication networks, transportation networks, etc. in particular in those type of networks which cannot be modelled into graphs but into Multigraphs.