An example of a slightly different kind -- not eliminating all calculation, but showing that "all calculations are easy" -- is Dehn's algorithmDehn's algorithm in combinatorial group theory. Dehn showed, using the combinatorics of hyperbolic tessellations, that the word problem for surface groups is solvable using only obvious word reductions.
In this case, calculation is avoided not so much by abstraction, but by thinking geometrically rather than algebraically.
