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?
Does Euler's solution of the Seven Bridges of Königsberg problem (considered by some the first theorem of graph theory) count?
From Morris Kline's Mathematical Thought from Ancient to Modern Times, Vol III, pg. 970:
In the paper containing George Cantor's final answer to the question of whether a function can have two different trigonometric series representations in the interval $[-\pi,\pi]$, Cantor "laid the foundation of the theory of point sets".
The Four Color Theorem. Who ever back then thought that we might use a computer to help us proving something...
Galois created groups and much more to attack and solve the problem of which polynomial equations are solvable by radicals.
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.
Cohen's proofs of the independence of the Axiom of Choice and the Continuum Hypothesis required the introduction of the notion of a generic set.
Fisk's short proof of Chvátal's art gallery theorem unexpectedly introduced graph coloring in a visibility problem in computational geometry.
Shelah's proof of the Whitehead problem and it independence. Until that point independence problems were considered as purely set-theoretical; but Shelah showed that this is not the case.