When does the use of afterburners save fuel?
If such a pair of states does exist it ought to be straddling a Mach boundary (I think)



e.g. A subsonic &amp; B trans-sonic. Or A trans-sonic &amp; B supersonic etc. 

Reasoning: To get from any point A to B in parameter space should usually be a state function i.e. path independent. 

i.e. You can give many tiny increments or one large increment.

In our context a parameter space means mostly a {velocity(v), altitude(h)} pair. 
To go from A{v1,h1} to B{v2,h2} you need a certain thrust profile. The required thrust profile is not unique. 

Now the above path independance statement is true in general, except when a dissipative force exists. Now you no longer have a state function. 

Since drag, a dissipative force, is indeed a strong function of speed it will help to not linger in the high drag region i.e. transsonic region with wave drag. In this zone the drag coefficient increases rapidly and you will have a total drag much higher than at both sub-sonic and super-sonic speeds.

And that would give a high thrust afterburner the advantage needed over a dry engine: On afterburner you can zoom through the transsonic region  rapidly than trudge through it incrementally on dry thrust. Thereby saving on drag losses. You are still consuming more fuel per unit time but the time spent is reduced so much that it makes the higher specific consumption worthwhile. 

Caveat: This is just an qualitative argument. A more mathematical argument is needed to add rigor to the analysis. 

Edit: Maybe these schematic plots help clarify my argument.