Neutrosophic set theory provides a new tool to handle the uncertainties in shortest path problem (SPP). This paper introduces the SPP from a source node to a destination node on a neutrosophic graph in which a positive neutrosophic number is assigned to each edge as its edge cost. We define this problem as neutrosophic shortest path problem (NSSPP). A simple algorithm is also introduced to solve the NSSPP. The proposed algorithm finds the neutrosophic shortest path (NSSP) and its corresponding neutrosophic shortest path length (NSSPL) between source node and destination node. Our proposed algorithm is also capable to find crisp shortest path length (CrSPL) of the corresponding neutrosophic shortest path length (NSSPL) which helps the decision maker to choose the shortest path easily. We also compare our proposed algorithm with some existing methods to show efficiency of our proposed algorithm. Finally, some numerical experiments are given to show the effectiveness and robustness of the new model. Numerical and graphical results demonstrate that the novel methods are superior to the existing method.