A colleague of mine has suggested the following solution:
The Dirichlet Distribution with N variables should be normalised by the following:
$\int_{0}^{\infty}d p_{1} (p_{1}^{x_{1}-1})\int_{0}^{\infty}d p_{2} (p_{2}^{x_{2}-1})\ldots \int_{0}^{\infty}d p_{N} (p_{N}^{x_{N}-1})\delta(1-\sum_{i} p_{i})$
In which $\delta(x)$ is the Dirac delta function. Note that as long as we restrict these integrals over a positive interval, we can extend them to $+\infty$ due to the $\delta$ function.
If we define the multi-variate beta-function to be the normalisation term of the Dirichlet distribution, then to prove the recursion relationship which is stated as its definition on wikipedia (and in effect, derive a more easy to evaluate form of the normalisation term), then we want to prove:
$\Gamma(\sum_{i} x_{i})\int_{0}^{\infty}d p_{1} (p_{1}^{x_{1}-1})\int_{0}^{\infty}d p_{2} (p_{2}^{x_{2}-1})\ldots \int_{0}^{\infty}d p_{N} (p_{N}^{x_{N}-1})\delta(1-\sum_{i} p_{i})= \prod _{i} \Gamma (x_{i})$
If we expand the gamma function on the LHS, and write the product as a "double" (in reality N+1) integral, the LHS =
$\int _{0}^{\infty}dzdp_{1}dp_{2}\ldots dp_{N}\hspace{3mm} e^{-z}\cdot z^{\sum_{i} x_{i} -1} \cdot p_{1}^{x_{1}-1}\ldots p_{N}^{X_{N}-1}\delta (1-\sum_{i}p_{1})$
Very similarly to Alex R's response above, we now perform the change of variables $(p_{1}, p_{2},\ldots, p_{n}, z) \to (u_{1}, u_{2},\ldots, u_{n}, y) $ using the transformation $p_{i}=\frac{u_{i}}{y}, z=y$
This results in a transformation Jacobian of $\frac{1}{y^{N}}$, and the limits of integration all remain $0 \to \infty$. Consequently, our multiple integral can be written as:
$\int _{0}^{\infty}du_{1}\ldots du_{N}dy \hspace{2mm}(\frac{1}{y^{N}})(\frac{u_{1}}{y})^{x_{1}-1}(\frac{u_{2}}{y})^{x_{2}-1} \ldots (\frac{u_{N}}{y})^{x_{N}-1}\delta(1-\sum_{i}\frac{u_{i}}{y})y^{\sum_{i}x_{i} -1}e^{-y} $
By writing the delta function as $y\cdot \delta (y- \sum_{i}u_{i})$, we can make all of the factors of y cancel, meaning we only need to evaluate the integral
$\int _{0}^{\infty}du_{1}\ldots du_{N}dy\hspace{2mm}u_{1}^{x_{1}-1}u_{2}^{x_{2}-1}\ldots u_{N}^{X_{N}-1}e^{-y}\delta(y-\sum_{i}u_{i})$
which after integrating against the delta function, does indeed give us
$\int_{0}^{\infty} du_{1}du_{2}\ldots du_{N}\hspace{2mm} u_{1}^{x_{1}-1}u_{2}^{x_{2}-1}\ldots u_{N}^{x_{N}-1}e^{-\sum_{i}u_{i}}=\prod_{i} \Gamma(x_{i})$
and hence the term which normalises the Dirichlet distribution is indeed equal to $\frac{\Gamma(x_{i})}{\Gamma(\sum_{i}x_{i})}$