Inflation models are most often studied using both classical and quantum treatment of the field ('inflaton field'), on a background of ordinary classical spacetime treated by General Relativity. We can then say the inflaton field provides a stress-energy tensor, and it is often taken to be homogenous and isotropic to first approximation, though this may be questionable. But if it is homogenous and isotropic then the Einstein field equation does indeed take the Friedman form. All we need is the equation of state of the inflaton field. This is provided by each inflation model, but broadly speaking they all feature something like a dark-energy contribution which dominates initially. So this gives rise to an exponential dependence of scale parameter on time. This is just what the Friedman equation (which is a way of writing the Einstein field equation) predicts for an equation of state like that of a cosmological constant.
What you are really asking about, then, is not so much a replacement for the Friedman equation as a way of specifying an equation of state which can treat the transition from inflation to more 'ordinary' evolution.
I have written this answer based on some general knowledge; I am not an expert on inflation so if an expert is willing to fill this out or correct it, then so much the better.