The aim of this work is to analyze and get the approximate analytical solutions for dropping shock response of cushion packaging system with tangent nonlinearity. The tangent system was simplified to the cubic-quintic nonlinear system to obtain the dimensionless dynamic equation after dimensionless treatment. The Newton-harmonic balancing method was used to obtain the first-order and the second-order analytical solutions of dropping shock response to the non-dimensional dynamic equation, and the analytical expressions of important parameters including the maximum displacement, the maximum acceleration of the system response and the dropping shock duration. The example analysis showed that the second order approximate solutions of the NHB method were very similar to those obtained from the Runge-Kutta method. The relative error was controlled within 2%. The newton-harmonic balancing method provides a new effective solution method for dropping shock response analysis of nonlinear packaging system.