Let us forget integrability questions for a moment and assume that $\mathrm{e}^{r|\xi|}$ is integrable for every positive $r$. Then the function $g$ is well defined and continuous everywhere.
First step: as you noted, $g(0)=1$ and g'(0)=E(\xi), hence g'(0)<0 and $g(r)<1$ for $r>0$, $r$ close to $0$.
Second step: assume that $P(\xi>0)\ne0$ (otherwise, as shown by Theo, the result is false), and choose $x>0$ such that $P(\xi\ge x)=p>0$. Then $\mathrm{e}^{r\xi}\ge\mathrm{e}^{rx}\mathbf{1}_{\{\xi\ge x\}}$ for every nonnegative $r$, hence $g(r)\ge\mathrm{e}^{rx}p$ for every nonnegative $r$, and one sees that $g(r)\to+\infty$ when $r\to+\infty$.
Conclusion: since the function $g$ is continuous, the intermediate value theorem implies that there exists a positive $r$ such that $g(r)=1$.
When $\mathrm{e}^{r|\xi|}$ is not integrable for at least some positive $r$, three different things may happen: $g(r)$ may be infinite except at $r=0$, or finite for every $0\le r and infinite for $r\ge r_0$, for a given finite $r_0$, or finite for every $0\le r\le r_0$ and infinite for $r>r_0$.
The conclusion that there exists a positive $r$ such that $g(r)=1$ would still hold if for example, there exists a positive $r_1$ such that $g(r_1)$ is finite and $g(r_1)>1$. It would fail, for example, if $g(r)$ is finite for every $0\le r\le r_0$ and infinite for $r>r_0$, with $g(r_0)<1$.