The theory of Feynman path centroid dynamics is applied to the calculation of quantum barrier crossing rates. The formulation starts from the exact definition of the quantum survival probability of the reactant state, and the reaction rate is then defined as the steady-state limit of the decay rate of the survival probability. A formulation is given in terms of exact centroid dynamics. Then, based on an approximation for the initial reactant state and the centroid molecular dynamics (CMD) approximation for the dynamics, a new approximate rate expression is obtained which is equal to the path integral quantum transition state theory (PI-QTST) expression multiplied by a transmission factor of order unity. This factor varies with the choice of the dividing surface in the low temperature limit, but it is invariant to that choice at higher temperatures. It is then shown that the PI-QTST rate expression results from the quadratic barrier approximation for the calculation of the transmission factor only. The potential to use the new rate expression as an improved version of the PI-QTST is also tested for model systems. For certain choices of the dividing surface, it is shown that the new reaction rate expression results in improvement over the PI-QTST results. The overall formulation also yields a better understanding of the barrier crossing dynamics viewed from the centroid perspective and the rigorous origin of the PI-QTST formula.