Example of a conjecture/theorem which required an entirely new idea to prove When Andrew Wiles proved Fermat's Last Theorem, he built upon ideas from elliptic curves which already existed. Is there an example of a conjecture/theorem which was proved using an unexpected argument?  

The premise of this question is dubious, insofar as it suggests that Wiles's work was less grounded in "entirely new idea[s]" than other major breakthroughs.
While some parts of the argument involved technical improvements on already known results, Wiles's argument (partly joint with Richard Taylor) also had many brilliantly new ideas in it (not surprisingly, since it solved the most well-known open problem in mathematics at the time), including an amazing criterion in commutative algebra for deducing that a homomorphism of rings is an isomorphism, and strikingly original arguments in the deformation theory of Galois representations.
Just to editorialize a little more: I think it's generally a mistake to think in terms of arguments being "entirely new" or representing a complete break from what went before.  In my experience, it's generally more profitable to explore the connections between important new mathematics and the mathematics that has come before.  Generally, there are many threads connecting the old and the new, and analyzing the pattern that they form can be inspiring and illuminating.
