Classical action as $S$-matrix generating functional at the tree-level

Question

I have seen repeated statements that the $S$-matrix generating functional at the tree-level is the same as the action calculated on the classical solutions of the equation of motion corresponding to this action. But I don't understand this statement. At least then, that the $S$-matrix generating functional takes into account the law of conservation of momentum. Is it possible to somehow rigorously prove this statement?

References:

1. "Introduction to functional methods" Faddeev.
2. https://doi.org/10.1103/PhysRevD.37.1485 formula (9)