The multiplicative perturbation method with precise matrix exponential computation is developed for the buckling analysis of axially compressed truncated conical shells (TCSs) that are commonly encountered in engineering. To overcome the limitation of conventional methods in terms of assuming solution forms, the multiplicative perturbation method is introduced to tackle the governing partial differential equations (PDEs) with variable coefficients. Specifically, the governing equation in matrix form for a buckled TCS is first formulated in the state space. The multiplicative perturbation method is then employed to convert the matrix differential equation with variable coefficients into the state transition equations with constant coefficients, in which the arisen matrix exponential is computed by the precise integral method. Finally, the state transition equations and the boundary conditions are integrated into an entire matrix equation, whose solution provides the buckling loads and buckling modes of the TCS. The convergence study and comprehensive numerical and graphic results are presented. Given the new solutions, the effects of some crucial size parameters as well as boundary conditions on the critical buckling loads are quantitatively studied. Due to the merits on solving PDEs with variable coefficients, the developed method may be extended to more intractable plate and shell problems.