Shortcut to equilibration (STE) refers to protocols that accelerate significantly the equilibration of a given system. Realizing such protocols is an interesting and relevant problem for enhancing the performance of quantum heat engines. However such a challenge comes often with the challenge of describing the dynamics of a driven open quantum system in a practical way for control. In particular in order to derive a master equation, one needs to evaluate the propagator of the closed dynamics, but this can be complicated for a general time-dependent Hamiltonian. To overcome this issue, we propose to use the dynamical invariant also known as the Lewis-Riesenfeld invariant. This LR invariant is an operator that can be used to solve the time-dependent dynamics of an isolated  quantum system. Here we use it to derive a time-dependent non-adiabatic master equation, and we find the STE protocol by reverse-engineering the equations of motion that we obtained. We focus on the isothermal stroke and illustrate the approach with the damped harmonic oscillator where we compare our shortcut with simple protocols (quench and smooth ramp) and also characterize the dynamics of the system during the STE.