In what follows we assume $\Re(a)>0$ and $\Re(b)>0$. Begin with the case $a+b=k\in\mathbb N$. Using Pochhammer contour $P$, one can relate what's going on on $[0,1]$ to what is going on on a circle $C:=x_*\mathbb S^1$, $|x_*|>1$. Indeed, looking carefully at determinations of $f(z):=z^{a-1}(1-z)^{b-1}$ one has $$ \oint_Pf(z)dz = (1-\exp 2ib\pi)\oint_C f(z)dz =(1-\exp 2ib\pi)(1-\exp 2ia\pi)\int_0^1f(z)dz ~~~(*)$$ the last equality being given by Pochhammer formula. Since $f$ is holomorphic near $\infty$ we have $$\oint_Cf(z)dz = -\oint f(1/x)\frac{dx}{x^2} , $$ which is a contour integral. It can be evaluated by looking a the expansion of $$(1-x)^{b-1} = \sum_n \frac{\Gamma(b)}{\Gamma(n+1)\Gamma(b-n)}x^n$$ The residue of $f(1/x)\frac{dx}{x^2}$ at $0$ is then $(-1)^b\frac{\Gamma(b)}{\Gamma(a+b)\Gamma(1-a)}$ which allow to conclude using Gamma reflection formula $\Gamma(1-a)\Gamma(a)\sin(a\pi)=\pi$.
The next step is to deal with the case $a+b=p/q\in\mathbb Q$, then conclude by density and continuity. This case is dealt with by taking a linear combination of $\oint_Cf(z)dz$ with weights $\exp (2in\pi/q)$ to obtain the same kind of relation as $(*)$. I'll write details later, but they should be straightforward.