Arguments by mathematical induction seem to provide an entire class of examples of the phenomenon, where computation is replaced by a higher level of reasoning.
To give a very elementary example, we all know how to use induction to prove that $1+2+\cdots+n=n(n+1)/2$. Thus, the comparatively abstract inductive argument predicts definite values for concrete sums $1+2+\cdots+105$. And surely mathematics is covered with dozens or hundreds of similar examples, of every degree of complexity. We understand the iterates of a function $f^n(x)$ without calculating them, or the powers of a matrix $A^n$, or the successive derivatives of a function, all without calculation, by understanding the inductive relationship in effect at each step.
The same phenomenon is exhibited in more abstract applications of induction. To prove a property of all finite graphs, all finite groups or whatever kind of structure you have, by induction on the size, or degree or rank or whatever you are inducting on, replaces a calculation of the property in the individual cases with an abstract understanding of how the property propogates to more complex objects.