The result is well-known in a general setting (for path-connected Hausdorff topological spaces), but the proof is no picnic.
I can imagine that for a connected and open subset of $\mathbb{R}^n$ the proof simplifies a lot. I have several ways in mind to proceed. But before spending time writing down all the details, I was wondering if anyone already thought about a proof and wants to share it (maybe following the typical proof used to prove path-connectedness by showing that a specific subset is clopen, keeping in mind that injectivity must be assured).
The statement to be proved is the following.
Let $\Omega\subseteq\mathbb{R}^n$ be an open and connected subset. For any $a,b\in\Omega$ with $a\neq b$, there exists a continuous and injective map $\gamma:[0,1]\to \Omega$ such that $\gamma(0)=a$ and $\gamma(1)=b$.
EDIT
What I really want to prove is the following.
Let $\Omega$ be an arbitrary subset of $\mathbb{R}^N$. Let $a,b\in\Omega$ with $a\neq b$, such that there exists a continuous and injective map $\gamma:[0,1]\to \Omega$ such that $\gamma(0)=a$ and $\gamma(1)=b$. Assume that $b$ is an interior point of $\Omega$. Then there exists an injective map that connects $a$ to any point $c$ in a neighborhood of $b$