Very recently, Atangana and Baleanu defined a novel arbitrary order derivative having a kernel of non-locality and non-singularity, known as derivative. We analyze a non-integer order Anthroponotic Leshmania Cutaneous (ACL) problem exploiting this novel derivative. We derive equilibria of the model and compute its threshold quantity, i.e. the so-called reproduction number. Conditions for the local stability of the no-disease as well as the disease endemic states are derived in terms of the threshold quantity. The qualitative analysis for solution of the proposed problem have derived with the aid of the theory of fixed point. We use the predictor corrector numerical approach to solve the proposed fractional order model for approximate solution. We also provide, the numerical simulations for each of the compartment of considered model at different fractional orders along with comparison with integer order to elaborate the importance of modern derivative. The fractional investigation shows that the non-integer order derivative is more realistic about the inner dynamics of the Leishmania model lying between integer order.